Coding Theory
and Algebraic Geometry Proceedings of the International Workshop
held in Luminy, France, June 17-21, 1991
Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Trang 2
ators
Ge ST A
Henning Stichtenoth Teawea de Lo cod a
Fachbereich 6 — Mathematik und Informatik = © aa =œSó65
Universitat GHS Essen
Universitatsstr 3, W-4300 Essen 1, Fed Rep of Germany Michael A Tsfasman
Institute of Information Transmission (IPPI) 19, Ermolovoi st., Moscow, GSP — 4, 101447, Russia
Mathematics Subject Classification (1991): 14-06, 94-06, 11-06
ISBN 3-540-55651-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-5565 1-6 Springer-Verlag New York Berlin Heidelberg
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Foreword
The workshop ” Algebraic Geometry and Coding Theory - 3” organized by the Insti- tute of Information Transmission (Moscow), University of Essen, Equipe Arithmétique
et Théorie de I'Iaformation de C.N.R.S (Marseille-Luminy), and Group d'Étude du
Codage de Toulon took place in the Centre International de Rencontres Mathematiques, June 17-21, 1991
The workshop was a continuation of AGCT-1 and AGCT-2 that took place in 1987
and 1989, respectively It is to be followed by AGCT-4 in 1993, etc., each time held in
C.LR.M
The list of participants follows
Trang 3Aubry, Yves (Marseille)
Blahut, Richard E (Owego, N.Y.) Boutot, Jean-Francois (Strasbourg) Bruen, Aiden (London, Ontario) Carral, Michel (Toulouse)
Chassé, Guy (Issy les Moulineaux) Cherdieu, Jean-Pierre (Guadeloupe) Cougnard, Jean (Besangon)
Deschamps, Mireille (Paris) Driencourt, Yves (Marseille) Duursma, Iwan M (Eindhoven) Ehrhard, Dirk (Diisseldorf) Gillot, Valérie (Toulon) Guillot, Ph (Genevilliers) Hansen, Johan P (Aarhus) Harari, Sami (Toulon)
Hassner, Martin (San Jose, Ca.) Helleseth, Tor (Bergen)
H¢gholdt, Tom (Lyngby) Katsman, Gregory (Moscou)
Kumar, Vijay (Los Angeles)
Kunyavskii, Boris E (Saratov) Lachaud, Gilles (Marseille) Langevin, Philippe (Toulon) Lax, Robert (Baton Rouge)
Le Brigand, Dominique (Paris)
Li, Winnie (Pennsylvania State)
Lopez, Bartolomé (Madrid) Luengo, Ignacio (Madrid) Michon, Jean-Francis (Paris) Munvera, Carlos (Valladolid) Nogin, Dimitri (Moscou)
Pedersen, Jens Peter (Lyngby)
Pellikaan, Ruud (Eindhoven) Perret, Marc (Marseille) Polemi, Despina (New York) Rodier, Francois (Paris) Rodriguez, M.-C (Madrid) Rolland, Robert (Marseille) Rotillon, Denis (Toulouse)
Seguin, Gerald (Kingston, Ontario) Serre, Jean Pierre (Paris)
Shahrouz, Henri (Cambridge, Ma.) Shparlinski, Igor (Moscou)
Skorobogatov, Alexei (Moscou) Smadja, René (Marseille) Sole, Patrick (Valbonne) Stichtenoth, Henning (Essen) Stokes, Philip (Valbonne) Thiongly, Augustin (Toulouse) Tsfasman, Mikhail (Moscou) Vladut, Serge (Moscou) Voss, Conny (Essen)
Wolfmann, Jacques (Toulon)
Contents
H Stichtenoth, M.A Tsfasman: Algebraic Geometry and Coding Theory An Introduction
Y Aubry: Reed-Muller Codes Associated to Projective Algebraic Varieties
D Ehrhard: Decoding Algebraic-Geometric Codes by Solving a Key Equation
G Frey, M Perret, H Stichtenoth: On the Different of Abelian Extensions of Global Fields
A Garcia, R Lax: Goppa Codes and Weierstrass Gaps
N Hamada, T Helleseth: On a Characterization
of Some Minihypers in PG(t, q) (q = 3 or 4) and its Applications to Error-Correcting Codes J.P Hansen: Deligne-Lusztig Varieties and Group Codes
G.L Katsman, M.A Tsfasman, S.G Viadut: Spectra of Linear Codes and Error Probability of
Decoding
P.V Kumar, K.Yang: On the True Minimum Distance of Hermitian Codes
B.E Kunyavskii: Sphere Packings Centered at S-units of Algebraic Tori
J.P Pedersen: A Function Field Related
Trang 4R Pellikaan: On the Gonality of Curves, Abundant Codes and Decoding
LE Shparlinksi, M.A Tsfasman, §.G Vladut: Curves with Many Points and Multiplication in
Finite Fields
P Stokes: The Domain of Covering Codes
M.A Tsfasman: Some Remarks on the
Asymptotic Number of Points C Voss: On the Weights of Trace
Codes
F Rodier: Minoration de Certaines Sommes
Exponentielles Binaires
A.N Skorobogatov: Linear Codes, Strata of Grassmannians, and the Problem of Segre 132 145 170 178 193 199 210 Algebraic Geometry and Coding Theory An Introduction
Henning Stichtenoth, Michael A Tsfasman
H.St.: Fachbereich 6 - Mathematik, Univ.GHS Essen, D-4300 Essen 1, Germany
M.Ts.: Institute of Information Transmission,
19 Ermolovoi st., Moscow GSP-4, U.S.S.R
About ten years ago V.D.Goppa discovered an amazing connection between the theory of algebraic curves over a finite field F, and the theory of error-correcting block
q-ary codes The idea is quite simple and generalizes the well known construction of Reed-Solomon codes The latter use polynomials in one variable over Fre and Goppa
generalized this idea using rational functions on an algebraic curve
Here is the definition of an algebraic geometric code (or a geometric Goppa code) Let X be an absolutely irreducible smooth projective algebraic curve of genus g over
F, Consider an (ordered) set P = {P,, , Pn} of distinct F,-rational points on X
and an F,-divisor D on X For simplicity let us assume that the support of D is disjoint
from P The linear space L(D) of rational functions on X associated to D yields the linear evaluation map
Bup : L(D) + F* ƒ¬>Œ(ŒP\) f(Pa))
The image of this map is the linear code C = (X,P, D)z we study
The parameters of such a code can be easily estimated Indeed, let P= P\+ +Pn,
then the dimension k is given by k=£(D) - (D —P) and in particular if 0 < deg D < n then " k =£(D) > deg D — g+1 The minimum distance d>n — deg D
since the number of zeroes of a function cannot be greater than the number of its poles We get the lower bound
k+ả>n+liT-g
which is by g worse than the simplest upper bound valid for any code
Trang 5An equivalent description of these codes can be given in terms of algebraic function
fields in one variable over F, The curve X corresponds to the function field F = Fy(X),
and F,-points on X correspond to places of F of degree one
Originally, Goppa used the dual construction using differentials on X rather than
functions, and the residue map
Unfortunately, there are at least two different traditions of notation The second
one uses D for our P and G for our D , and the code is denoted C,(D, G)
The construction can be generalized in several directions In particular one can use sheaves (or some other tricks) to avoid the condition PNSuppD = 9 The generalization to the case of higher dimensional algebraic varieties looks very promising but so far the results are few
There are several main streams of the development of the theory Let us briefly — discuss some of them
' Asymptotic problems One of the fundamental problems of coding theory is
to construct long codes with good parameters (rate and relative minimum distance)
One of the starting points of the theory was the construction of long codes which are asymptotically better than the Gilbert-Varshamov bound The other asymptotic question is which codes can be constructed in polynomial time
Specific curves There are many interesting examples of curves with many F,- points which lead to codes with good parameters Sometimes such curves and codes have nice additional properties, such as large automorphism groups
Spectra and duality The study of weight distribution and of duality leads to interesting questions of algebraic geometry, such as the study of Weierstrass points and special divisors on a curve
Decoding Surprisingly enough the decoding problem can be set in purely alge-
braic geometric terms and again one needs information about special divisors
Exponential sums Another component of the picture is the theory of exponential sums closely related both to algebraic geometry and to coding theory
Related areas The theory of algebraic geometric codes has either analogues
or applications in several other topics Such are sphere packings and spherical codes, multiplication complexity in finite fields, graph theory, and so on These applications also require subtle information about the geometry and arithmetic both of function fields and of number fields
To conclude, the first ten years of development show that the connection between
algebraic geometry and coding theory proves fruitful for both, giving new results and posing many exciting questions
ration The papers are too numerous to list them here and we refer to the extensive
bibliography in [Ts/V1] and to references given in the papers of this volume Here is the list of books
[Go] V.D.Goppa, Geometry and Codes Kluwer Acad Publ., 1988
[Mo] C.J Moreno, Curves over Finite Fields Cambridge Univ Press, 1991
[Sti] H.Stichtenoth, Algebraic Function Fields and Codes Springer-Verlag (in preparation)
[Ts/VI| M.A.Tsfasman, S.G.Vladut, Algebraic Geometric Codes Kluwer Acad Publ., 1991
[vG/vL] J.H.van Lint, G.van der Geer, Linear Codes and Algebraic Curves
Birkhauser, 1988
Trang 6Projective Algebraic Varieties
Yves AUBRY
Equipe CNRS "Arithmétique et Théorie de I'Information" C.LR.M Luminy Case 916 - 13288 Marseille Cedex 9 - France
Abstract
The classical generalized Reed-Muller codes introduced by Kasami, Lin and Peterson [5], and studied also by Delsarte, Goethals and Mac Williams [2], are defined over the affine space A"Œ@ over the finite field Fy with q elements Moreover Lachaud 6], following Manin and Viadut [7], has considered projective Reed-Muller
codes, i.e defined over the projective space P"(F,)
In this paper, the evaluation of the forms with coefficients in the finite field Fy is made on the points of a projective algebraic variety V over the projective space P°Œ,) Firstly, we consider the case where V is a quadric hypersurface, singular or not, Parabolic, Hyperbolic or Elliptic Some results about the number of points
in a (possibly degenerate) quadric and in the hyperplane sections are given, and also is given an upper bound of
the number of points in the intersection of two quadrics
in application of these results, we obtain Reed-Muller codes of order 1 associated to quadrics with three weights and we give their parameters, as well as Reed-Muller codes of order 2 with their parameters
Secondly, we take V as a hypersurface, which is the union of hyperplanes containing a linear variety of codimension 2 (these hypersurfaces reach the Serre bound) If V is of degree h, we give parameters of Reed- Muller codes of order d < h, associated to V
1 Construction of the Projective Reed-Muller codes
We denote by P"Œq) the projective space of dimension n over the finite field Fy with q elements, q a power of a prime p The number of (rational) points (over Fy) of P"Œq) is:
n+1
Hạ =IPSŒQ)I=q” + q + +ạ+.1=E TS,
The family { W; }9.;<, is clearly a partition of P"(F,)
Let FqlXo, Xy, Xali be the vector space of homogeneous polynomials of degree d with (n+1) variables and with coefficients in Fy Let V be a projective algebraic variety of P°ŒQ) and let | V | denotes the number of theirs rational points over Fy Following G Lachaud ((6]), we define the projective Reed-Muller code R(d,V) of order d associated to the variety V as the image of the linear map 2 FqlXo, X1, os Xaly > Fl defined by c(P) = ( C,(P) )xev, where P(xo, ., Xp) : ox(P) = =e a) if x = (Kyi xe Wis 1
G Lachaud has considered in [6] the case where V = P"Œ), with d < q.Moreover, A.B Sorensen has considered in [12] the case where V is equal to P"ŒQ) too, but with a weaker hypothesis on d
Now we are going, firstly, to study the case where V is a quadric, degenerate or not, but before we have to establish results on quadrics and this is the subject of the following paragraph
2 Results on quadrics
In what follows the characteristic of the field Fy is supposed to be arbitrary (the results hold in characteristic 2 as well as in characteristic different of 2)
2.1, The quadrics in P"(Fq)
In this paragraph, we recall some properties of quadrics in the projective space P"ŒQ) J.F Primrose has given in [8] the number of points in a nondegenerate quadric (see below the definition of the rank of a quadric), and D.K Ray-Chaudhuri [9] gave more general results (which with, in a particular case, we recover those of Primrose's) We are going here to follow the notations of J.W.P Hirschfeld in [4]
A quadric Q of P"(F,) is the set of zeros in P"(Fq) of a quadratic form —
Trang 7coordinates oh which, —— ‘mute the form F ith a fewer number of variables More precisely, if T is an invertible linear transformation defined over PX, q) denote by Fr@@) the form F(TX) Let i(F) be the number of indeterminates appearing explicitly in F The rank r(F) of F (and by abuse of language, of the quadric Q), is defined by :
x rự) = mịn in)
where T ranges over all the invertible transformations defined over F, A form F (and by abuse the quadric Q) is said to be degenerate if
r(F) <n+1 erwise the quadric are nondegenerate
Cetus ronan hota ne degenerate if and only if it is singular (see [4])
We recall after J.W.P Hirschfeld (see [4]) that in P"(Fy), the number of different types of nondegenerate quadrics Q is 1 or 2 as n is even or odd, and they are respectively called Parabolic (®, and Hyperbolic (A) or Elliptic (£)
The maximum dimension g(Q) of linear subspaces lying on the nondegenerate quadric Q is called the projective index of Q The projective index has the following values (see [4]) : — -Í n- 3 g@= 72, g0)= 7ˆ, sœ=s The character «(Q) of a nondegenerate quadric Q of P"(F,) is defined by : o(Q) = 2g(Q) —n + 3 Consequently, we have :
OM=1, O)=2, a2) =0
Then, we have the following proposition (for a proof see [4]) :
Proposition 1 : The number of points of a nondegenerate quadric Qof P"ŒQ) 1S;
IQI=n_ 1 + (@(Q) — 1) gia 1⁄2,
We want now to evaluate the number of points of a degenerate quadric Q = Z(F) of P"ŒQ) of
rank r (called a "cone" of rank r) ¬
We have the following decomposition in disjoint union (an analogous decomposition 1s given
by R.A Games in [3]) :
Q=Va-1rYVQy-1-
We have set -
_ Vn_r=({(0:0: :0:yy: :yn)€ PˆŒạ) =P"~"(Fy), The
if we suppose that the r variables appearing in the quadratic form F are Xọ, XỊ, Ấy _ 1 set V, _, is called the vertex of Q, and is the set of singular points of Q We note also
Q”7_1=[ŒQ: :Xy_1:Yy: : Yn) € PF | F(Xg, Yn) = 0 and the x; are not all zero} Let Q,_ 1 be the nondegenerate quadric of P*~ (F,) associated to Q, i.e defined by
Qe 1 = Zpy — 1 (Fy 1
Qe 1 =E (Xo: - Xp 1) © PYM) LF 1x0, ., xe) =0 },
where F, _ 1(Xp, , Xy_ 1) = F(X, »X,) The (degenerate) quadric Q will abusively be said to be parabolic, hyperbolic or elliptic according to the type of its associated nondegenerate
quadric Q, _ Its character «(Q) is by definition the character @(Q;_ 1) of Qr_ ¡
Then, we have the following result which can be found in R.A Games [3]:
Theorem 1 : The number of points of a quadric Q of P'F) of rank ris :
1QU= tq _ 1 + (@(Q)— 1) q0 ~1/2
and we have œ(Q) = 1 if ris odd, and œ(Q) = 0 or œ(Q) = 2 ïfr is even
In particular, a quadric of odd rank is necessarily parabolic, and a quadric of even rank is
hyperbolic or elliptic ,
Corollary : Let Q be a quadric of PX» with n 2 2, We have :
M-2S1Qisay_,+q"-!,
and the bounds are reached
Observe that the lower bound is the Warning bound and that the upper bound reaches the following Serre bound, conjectured by Tsfasman, which says that (see (11) if Fe Fq[Xo, Xnlg is a nonzero form of degree d < q, with n > 2, then the number N of zeros of F in F," is such that:
N <dq™~1_@-1) q?-2,
2.2 Hyperplane sections of quadrics
This paragraph deals with the number of points in the intersection of a quadric anda ' hyperplane When the quadric is nondegenerate, the result is known (see for example [13]) R.A Games has given the result when the quadric has the size of a hyperplane, provided the quadric itself is not a hyperplane (see (3}) Furthermore, ILM Chakravarti in [1] has solved the case when the quadric is 1-degenerate, that is a quadric of rank n in PˆŒ,)
We are going, here, to consider the general case, i.e quadrics in P"ŒQ) of any rank
Trang 8if Q is hyperbolic or elliptic (indeed r(Q) is necessarily even, and if H is not tangent we have r(Q 7 H) = 1(Q) — 1 hence odd, then Q 1 H is parabolic) ; and QQH becomes hyperbolic or elliptic if Q is parabolic (same reason rest on the parity of the ranks)
Now if the hyperplane H is tangent to Q, we have the following proposition (see [13]) : Proposition 2 : The quadric Q 7 H is of the same type as the nondegenerate quadric Q if the hyperplane H is tangent to Q
Then, we can give the result about the hyperplane sections of a quadric of any rank :
Theorem 2 : Let Q be a quadric of P°"ŒQ) of rank r whose decomposition is * Q=Va-1rVQr-1 and let H be a hyperplane of P°Œ,) Then : aIfHDV,., then IQHI=1g_z+(@(Q;_— 1 Hạ) — 1) qn ~r~ 1⁄2 if H, is not tangent to Q ¡› and QHI=1a_2 + (6(Q)— 1) qn~?2 if H, is tangent to Q,_ 1, where H, is the hyperplane of P'.~!ŒQ) defined by Hy = Zpe (0)
where h is the linear form in Fg[Xo, X,_ ie defining H ; moreover w(Q,_ 1 0 H,) is equal to 1 if Q is hyperbolic or elliptic, and equal to 0 or 2 if Q is parabolic b) If HD Va_, then IQSHI=1a_2 + (@(Q) — 1) qÊn ~r~2/2, Prooƒ : We suppose that the r variables appearing in the quadratic form F defining Q are Xp.X Xr _ 1: If we set H; the hyperplane whose equation is X; = 0, we have Vn_r= Họ O Hị 3 Hr_ - But * : Q¬aH=(Vạ_rcQ r-ÐOH=(Vạ_rOH) 2(QÌ;_¡H), Thus IQOHI=1 Vg OHI +1 Qy-1 OHI ~ 1Va-1 0 Q-1 OHI; butVạ_rQ r-1 =@, thus: | -IQnaHI=lVa_rSHI+iQ);_¡HI Furthermore, the linear form h defining H is such that h e FqiXo Xr_ il Indeed, if a _h= YaN¡, 0
we have for all ¡ >r, P¡= (0: :0: 1 :0: : 0) where the 1 is at the i" - coordinate,
Địc Vnạ_ and HĐ Vn _r thus h(P;) = 0 But h(P;) = aj , thus a; = 0 for all i2 r Hence, IQy_iOHI=g°TFf1IQ —raH,L — The quadric Q;_¡H, of P'~2ŒQ) is degenerate or not, according as H, is tangent or not to Qr_¡ Now: 7 If H, is not tangent to Q,_, , then by proposition 1, (since Qr_¡n H, ¡s nondegenerate in PT ?®ŒQ) ), we have : ` 1Qp_1 VHy! = m3 + (OQ 1 Hạ) — D) q~ 392, Thus
IQaHI=n_r+g?T7* TQ, ¡e3 Hạ Í= n2 + (@(Q,_ ¡ c3 Hạ) — 1) qU-T~ 9,
— If H, is tangent to Q,_ 1 , then by theorem 1, we have :
IQr OH, 1 = m3 + (O(Q_1 A H,)— 1) gt 2,
but by proposition 2 we know that @(Q, _ 1 © Hy) = @(Q, _ 1),which is equal to @(Q) by
definition Finally, ,
IQAHl=mm_p+q"-**! (m_3 + (@Q—1) qt)
_ =n_2 + (@(Q)— 1) qn—2/2, - 2°) Suppose now that H not contains Vn_r
Wehave Vạ_rH=Họạn Hị O aHr_¡H, thu IVa_rHI=lPR~T~ A)L=n r1 n If h= zak is the linear form defining H, there exist necessarily one j, r $j <n, such that a; #0 Thus
Qr_¡¬H=({ G: ôTT tYp? ôâjS1?E!j +1! «« tYn) 6 PPŒQ)
with Q¿ _ ¡(Xo, Xy_ ¡) = O and the x; are not all zero }, where t is such that
ajt = — â0X0 — — ấy _ 1Xy — 1 — ÂrŸr— «« — Bị ~ 1Ÿ] — 1 — Bj 4 1] 1 + — ADYn-
Thus
IQ2_iaHI=qgf=r+Ð=liQ,_rị
Trang 91Q"-1 AHI =4""" (my + ((Q_1)- Dat?)
and finally :
-_!QOaHI=im_r~i+?T(p~2 + (@(Qr— 0)— 1) qữ~®/Z)
= Ta _2 + (@(Q) — 1) qiên ~r~ 2/2,
which concludes the proof.¢
2.3 Intersection of two quadrics in P"(F a”
The subject matter of this paragraph is to estimate the number of points in the intersection of two quadrics in P"ŒQ) with n > 1 We give an exact value of this number in a particular case, and an upper bound in the general case (Theorem 3), inspired by an another upper bound of W.M Schmidt ([10] p.152) We need first a lemma :
Lemma : If Q, and Qy are two distinct quadrics in P'(Fy), then :
IQiaQ21<ãn_¡+dh~Ê,
Proof : By theorem 1,1 Qy | = 7,_ 1 + (@(Q;)- 1) q?@"-? if ris the rank of Q) Thus : — ifr 2 4, we have 2m! <n —2 and then! Qy lSãn_Ị +qn~2 ; hence a fortiori
IQ¡Qzl<ãa_¡+q?~2,
—ifr=3 orr=1 then Q, is parabolic and
IQi¬Q¿l<IQ11=Za_¡<1a_¡+q?~2,
— ifr =2: either Q, is elliptic, and then! Q, |=, _,—q"~ ! and the result holds ; or
Q is hyperbolic, and then Q, is the union of two distinct hyperplanes We can suppose that the quadric Q, is also hyperbolic of rank 2, otherwise the same reasoning which we have made to
Q; must hold for Qo
We set Q] = Hp U Hy and Q») =H) U H,, and without loss of generality, we can take for Hj the hyperplane X; = 0 Since, by hypothesis, the quadrics Q, and Q» are distincts, two cases
can appear : * - » +
1°) The four hyperplanes are distincts, i.e t is different of 0 and 1 We obtain, simply in “counting” the points :
IQiaQzl=1a_4+4q*~”<a_¡ +q~ the preceding inequality is equivalent to (q — 1)2 > 0)
( Poe) Qy ‘and Q> have ao common hyperplane, i.e t= 0 or t = 1 Suppose that t = 0 Then,
we have : n
QịnQ¿=[ (0:xị: : xa) € PF) )U{( :0:0: x3: : xa) 6 PˆŒQ) ),
where the union is disjoint Hence :
IQi¬AQai=na_r +an~2 , and the upper bound of this lemma is reached in this case ¢
2
Theorem 3 : Let F)(Xp, ,.X,) and Fo(Xp, ,.X_) be two non zero quadratic forms with
coefficients in F, and let Qy and Q) respectively the two associated quadrics of P"Τ) Three
cases can appear :
1°) the forms F, and F¿ are proportional (i.e there exists Xe Fy such that F, = AF, ) and then: IQi¬Q¿i=lIQ¡[<lQ¿[ 2°) F¡ and Fy have a common factor of degree 1, and then : LQiAQ21=ãn_¡ +qh~2, 3”) F and Fy have no common factor (no constant), and then : -1 n-2 IQiAQ21Sã,-2+a ST - Ê8-—T (for q 27 this upper bound is indeed better than the lemma) Proof : 1°) Trivial `
2°) We are necessarily in the case where Q; and Q, are the union of two hyperplanes with one in common ; it is proved in the lemma
3°) Let F, and F2 be two quadratic forms without nonconstant common factor
The result is obvious if q < 4 Indeed, by the lemma, we have :
IQ AQIS m_1+q"~?
and furthermore,
n-1 n-2
Tn—1 +ah^?Sm— 2474 TT— - is equivalent to q <5
Suppose now that q > 4
We set, fori equal 1 and 2: ,
F’i(Xo, Xp) = F,(Xo › Xi+ctXo › X¿†c2Xo peeey XntcnXạ) = P¡(C1;€2, ,Cn) Xã +
The polynomials Pq and P¿ are not the zero polynomial (otherwise F and F¿ would be too), and are not also identically zero, since they have degree at most 2, and q > 4 implies that Fị and
Fy have at most 2q"~1 <q" zeros in F," (because a polynomial of degree d in F,[X} Xql
have at most dq™~ !zeros in Fạ" , See for example [10])
Moreover, the total number of zeros of P| added to those of P2 is then at most 4 qñ -I
which is < q" since q > 4
Thus it is possible to choose (C}, ;Cn) € F," such that
PỊ(C ,Cn) #0 and P2(C1› Cn) # 0
Thus, after a nonsingular linear transformation and after divided by P(cj, ,c,) and P2(C) Cn) respectively, we may Suppose without loss of generality that :
FIOXo, Xn) = Xổ + Xo gIỢK, Xn) + g2(X Xn) and
Trang 10where gị,hịe Fq[Xị Xa]) and.ga,h2e F,[Xụ, Xạ])
If we look at now the polynomials F, and Fy as polynomials in Xp, their resultant is a
homogeneous polynomial R(Xj, X,) of degree 4 By the well known properties of the
resultant, we can say that for any common zero (in Re ) (XeX, Xn) Of F1(Xọ, ,Xn) and
Fo(Xo, Xp,), we have R(x;, ,.X_) = 0
If we apply the Serre bound (see § 2.1) to the resultant R, we obtain that
the number of zeros in Fg" of R(Xj, Xp) is <4q"~1 ~ 3 q"~2,
Moreover, for such n-uple , the number of possibilities for xg is at most 2, and the forms Fy
and F» are of degree 2, thus the total number of common zeros (xg, ,X,) of Fy and F¿ in F,"*! is <8q?-! - 6qR~2, : And by the following usual equality : NẠŒ) = 1 +(q— 1) NpŒ) where NẠ (F) represent the number of zeros in A"'£) = Fl of F and Np(F) the number of zeros in P"ŒQ) of F, we deduce : : : 8qạ"-Ì _ @qn-2 _ 1 1Q,9 Q2Is q = 1 n-2 n-1 7 qi-l 6 n-2 =y,_2+6q +f — ï =fn-2tg- —T + Ta —T:
3 Projective Reed-Muller codes of order 1 associated to a quadric
Let Q be a quadric in P'(F,) of rank r, decomposing in disjoint union of its vertex V,, _ „ and of Q"- yp» where Q, _ ; is the nondegenerate associated quadric of P*~ 1) We will apply the results of § 2.2 to determine the parameters of the projective Reed-Muller codes of order 1 associated to Q Since these parameters vary according to the type of the quadric Q, we have to distinguish three cases
Theorem 4 (parabolic case) : Let Q be a parabolic quadric of PXF,) of rank r # 1 Then the projective Reed-Muller code of order 1 associated to Q is a code with three weights :
WỊ =q?~1_ qÉứn-r~ Ù/2 , wa= q1~ 1, qứn-r~ D2 W3 =qn-l
with the following parameters :
length = ny _ 1, dimension = n + 1, distance = q?~} — g@a-t- DA,
Theorem 5 (hyperbolic case) : Let Q be an hyperbolic quadric of P"ŒQ) of rank r Then the projective Reed-Muller code of order 1 associated to Q is a code with three weights :
wị= qh 1 + g@a- 0/2 | w2= q"~ 1 w3= qh~ 1, qên - 9/2 _ g@n-1-2)/2 with the following parameters :
length =n, _ 1 +q@"-"/?, dimension = n + 1, đistance = qñ — 1,
Theorem 6 (elliptic case) : Let Q be an elliptic quadric of PˆŒQ) of rank r > 2 Then the projective Reed-Muller code of order 1 associated to Q is a code with three weights :
w= qˆ~ 1 _ qin- 1⁄2, w2=q"~ 1 „ W3 = qi} _ qứn -9⁄2 + q(2n~ r~ 2/2
with the following parameters :
length = x,_, — q@°-/, dimension = n + 1, distance = q?~! — (29-2 |
Let us remark that we recover the results of J Wolfmann as a particular case of these results (see [13]), indeed he had considered the case of nondegenerate quadrics : his results correspond to the case where the rank r = n+1 Note that, here, the case Hf V,_, is excluded, and then we find only two weights for the hyperbolic and elliptic quadrics, but still three weights for the parabolic one We recover also the results of I.M Chakravarti (see [1]) : it corresponds to the case where the rank r=n
Proof : The lengths of the respective codes are equal to the number of points of the respective quadrics : theorem 1 gives the result ~
The map ¢ defining the code ( see § 1 ) is one to one, and thus the dimension of the code is equal to the dimension of F4[Xo. -Xaly over Fy, i.e.n + 1: indeed, if H is a hyperplane of P"ŒQ), ( which amounts to taking a linear form of FqlXo Xa] ), it is sufficient to apply the
results of Theorem 2 to see that | Q ¬A HÍ <l QI, and to have also the different weights ¢
4 Projective Reed-Muller codes of order 2 associated to a quadric
The map c: Fa[Xo. -Xnlo ~ F,2 as introduced in § 1 defining the projective Reed-Muller code of order 2 associated to the quadric Q has for domain the vector space of quadratic forms over Fy ; this is why we gave previously some results on the intersection of two quadrics of
P'(F,)
Theorem 7 (parabolic case) : Let Q be a parabolic quadric in PX\F,), n 2 2 If q 2 8 then the projective Reed-Muller code of order 2 associated to Q has the following parameters :
n-1
length = 1, _ 1, dimension = nat) distance 2 q'~! — 6 q~?- as
Theorem 8 (elliptic case) : Let Q be an elliptic quadric in P'(Fy) of rank r > 2 If q > 8 then the projective Reed-Muller code of order 2 associated to Q has the following parameters :
Trang 11We reserve the case where the quadric is hyperbolic of rank 2 for the theorem 10 (we have indeed more precise results)
Theorem 9 (hyperbolic case of rank r 2 4) : Let Q be an hyperbolic quadric in P'(Fy) of rank r24 If q 2 8 then the projective Reed-Muller code of order 2 associated to Q has the following parameters :
length = %,_ 4+ qán ~ 1/2, dimension =—.- ›
n-1
distance 2 q"~ Lạ qán - 1⁄2 _6qœ -„ 1 —r
Let us remark that we can have, for the theorem 9, the same results with a weaker hypothesis on q when the rank of Q is equal to 4 or 6, namely q > 5
Now we consider the case of maximal quadrics, that is hyperbolic quadrics of rank 2 By the corollary of theorem 1, the number of points of these quadrics reaches the maximum number of points of a quadric, and it is in this sense that we call them "maximal" We can remark that they are particular quadrics (they are the union of two distinct hyperplanes) The codes which are associated to them have a minimum distance precisely known These codes will have a generalization in the next paragraph
Theorem 10 (hyperbolic case of rank = 2) + Let Q be an hyperbolic quadric in P"ŒQ of rank 2 The projective Reed-Muller code of order 2 associated to Q has the following parameters :
length = ty _ y+ qa}, dimension =O +3) | distance = q"~2 (q~ 1)
Proof : The length of the codes is the number of points of the quadric Q, and is given by Theorem 1
Let Fe Fa[Xq, Xalp and Q’ = Z,9(F), Q=Z,a(F)
Either F and F are proportional, and then Q = Q’ Remark that there isq ~ 1 such non zero forms F ; thus there is at least q quadratic forms vanishing in Q, hence in the kernel of the map ¢ defining these codes We claim that there are no other forms in Ker(c), and thus the dimension of this codes is : Fa[Xo Xa]) dim(Im c) = dim “Keo "?- 2+ DO +2) _ tog (1Ker(e)1) _ (2+ 142) _ 4 _ nt+3n _ n(n 43) = 2 ny nn , Indeed, suppose now that F and F' are not proportional, we have by Theorem 3 : n-1 n-2 IQNQ I< mq_2 +225 - #9 ~ if Q is parabolic (Th 7), we have n—i n~ 2 Eạ~2+ TT - S1 -T—-<lQIq2~ 8q+6>0«>q>8
Moreover, F and F' cannot have a common factor of degree 1 since Q would be the union of two hyperplanes and thus would be hyperbolic
The minimum distance follows from the same inequality of the Theorem 3
~ if Q is elliptic (Th 8), F and F' cannot also have a common factor of degree 1, and we have :
n-1 n-2
Ty 24 —T ~ $9 <1Qt= mot - qin -19/2 if and only if q > 8 for r = 4, and
thus a fortiori for r 2 4, i.e since r is even, r > 2
~ if Q is hyperbolic of rank 2 4 (Th 9), the same reasoning gives a fortiori the results (indeed the hypothesis q 2 8 holds for more “smallest " quadrics)
~ if Q is hyperbolic of rank = 2 (Th 10):
* either F and F' have a common factor of degree 1, and by the Theorem 3 :
IQnQ?I=zn_ ¡+ qg”~2 which is < | QÌ = ạ_ ¡+ q8—,
+ or F and F' have not a common factor of degree 1, and by the lemma preceding Theorem 3
we have :1Q.0Q’1S%,_3+q"~2 which is <I.QI
The minimum distance in this case is :
1QU — (my_ytqh~?) = qt! ~ g?~? = q"-2(q- 1)
5 Projective Reed-Muller codes associated to a maximal hypersurface
We consider here hypersurfaces of degree h < q reaching the Serre bound, i.e which are the
union of h distinct hyperplanes containing a linear variety of codimension 2 The Serre bound
enunciated in § 2.1 has the following projective version : if F is a non zero form of degree h <q of FqXo Xa], then LZyŒ)1S.n_2+hq"— The construction of such varieties (called maximal) is easy ; indeed we can take for example : F= [] (Xo — AiX1) l<i<
where the A, are h distinct elements of F, We are going to construct projective Reed-Muller codes associated to such varieties
Theorem 11: Let V = ZpnŒ) be a variety of P"ŒQ) which is the union of h distinct
hyperplanes containing a linear variety of codimension 2, with h < q Then the projective Reed- Muller code of order d < h associated to V has the following parameters :
Trang 12Let us remark that we find again the projective Reed-Muller codes of order 1 associated to a maximal quadric (in the particular case h = 2 and d = 1)
Proof : The length of the code is equal to the number of points of the variety V which is, by
construction,
- Tn_2+ hqn~1,
The map c: FalXo»-Xnlf > ru defining the code is obviously one to one since d <h Thus the dimension of the code is equal to the dimension, over F q> of
FglX, Xql9 ie (7 F 9)
If V = Hy U U Hy then the subvariety V' of degree d of V defined by V’ = H, VU U Hy where the d hyperplanes are taken among the h defining V, is such that :
IV'l=a,_9+dq?-
Thus the minimum distance of the code is equal to :
IVI = (Tn_++dq?~Đ=ng?T1— qạ0=l=(n—đ)q0—Ì @
We can say more if we consider the particular case of the codes above of order 1 Indeed, it is easy to see that the hyperplane sections of such maximal varieties have three possible sizes,
nam€Ìy Ttn _ 1 ; n _ 2 OF Tạ _ 3 +hq^~ 2, Thus, the projective Reed-Muller code of order 1
associated to V (with h > 1) is a code with three weights :
wy=(h — 1)q™~!, wo=hq®~! , wz =hq?-!4+(1 — h)q?-2
and with the following parameters :
length = %,_2+hq"~!, dimension =n +1, distance = (h— 1) qnT1,
References
1 Chakravarti I.M., Families of codes with few distinct weights from singular and non-singular hermitian varieties and quadrics in projective geometries and Hadamard difference sets and designs associated with two-weights codes, Coding Theory and Design Theory - Part I : Coding Theory IMA vol 20
BI Delsarte P., Goethals J.M and Mac Williams F.J., On generalized Reed-Muller codes and their relatives, Inform and Control 16 (1970) 403-442
1 Games R.A , The Geometry of Quadrics and Correlations of sequences, TEEE Transactions on Information Theory Vol IT-32, No 3, May 1986, 423-426 [4] [5] [6] 1 [8] 19] [10] HH [12] H3] Hirschfeld J.W.P., Projective Geometries over Finite Fields, Clarendon Press, Oxford, 1979
Kasami T., Lin S and Peterson W.W., New generalization of the Reed-Muller codes - Part I : Primitive codes, IEEE Trans Information Theory IT-14 (1968),
189-199,
Lachaud G., The parameters of projective Reed-Muller codes, Discrete Mathematics 81 (1990), 217-221
Manin Yu.I and Vladut S.G., Linear codes and modular curves, Itogi Nauki i Tekhniki 25 (1984) 209-257 J Soviet Math 30 (1985) 2611-2643
Primrose E.J.F., Quadrics in finite geometries, Proc Camb Phil Soc., 47 (1951), 299-304
Ray-Chaudhuri D.K., Some results on quadrics in finite projective geometry based on Galois fields, Can J Math., vol 14, (1962), 129-138
Schmidt W.M., Equations over Finite Fields An Elementary Approach, Lecture Notes in Maths 536 (1975)
Serre, J.-P., Lettre 2 M Tsfasman, 24 juillet 1989, Journées Arithmétiques de Luminy, Astérisque, S.M.F., Paris, to appear
Sorensen A.B., Projective Reed-Muller codes, to appear
Wolfmann J., Codes projectifs a deux ou trois poids associés aux hyperquadriques
Trang 13Decoding Algebraic-Geometric Codes
by solving a key equation
Dirk Ehrhard*
1 Introduction
The recent work about the problem of decoding Algebraic-Geometric Codes has led to an
algorithm (e.g., see [2,3]) Another algorithm has been given by Porter, see [6,7,8,9,10],
generalizing Berlekamp’s decoding algorithm The main step is to solve a so-called “key-equation” For this purpose, Porter gave a generalization of Euclid’s algorithm for functions on curves Unfortunately, therefore he had to impose some strong restrictions to the code and its underlying curve, such that the resulting algorithm works only for a very small class of Algebraic-Geometric Codes Recently, the generalized Euclidian
algorithm was investigated and corrected by Porter, Shen and Pellikaan ({11]) and Shen
({12))
Here, we will show how to generalize Porters ideas to all Algebraic-Geometric Codes and moreover, how to solve the key equation by simple linear algebra operations Two observations on Porter’s methods have motivated our work:
1 The operations done by Porters algorithm at the so-called “resultant-matrix”, may be considered as a Gaussian algorithm, applied to the transposed matrix 2 The key equation may be viewed as linear
The result is given in section 2: A decoding algorithm of complexity order O(n’), that corrects up to li —1-—g)j errors, exactly as the well known algorithm does In section 3 we describe how strongly both algorithms are connected We conclude with section 4, giving a short overview over Porters algorithm and explaining, in what manner Porters work embeds in ours
2 The decoding procedure
2.1 The code
We will use the notations of [1]: Let X be a curve of genus g (i.e a non-singular, abso-
lutely irreducible projective curve defined over the finite field F,), Py, ,P, rational
“The author is with the Mathematisches Institut IV der Heinrich~-Heine-Universitat, 4000 Diisseldorf 1, Germany The contents of this paper are also part of the author’s Ph D Thesis ((13])
points on X, and G a divisor which has support disjoint from the P,’s We will as- sume that 2g — 2 < degG ¢ n+g—1 and define D := P, + +P, Then the code
C = C*(D,G) is the image of the linear, injective map Resp: Q(G-D) — Fj}
n r+ (Resp,y, ,Resp,7)
This is a linear [n, k, d], ~ code with k > n—1—deg G+g and d > d* = deg G —(2g —2) For details see [1] Note, that Resp may be extended to 2(X) in a canonical way
2.2 A theorem for preparation
Let G" be a divisor with support disjoint from the P,’s such that G—G’ is effective and
dim L(G’) = 0 An easy consequence is: 2(G — D) c 2(G' — D), which we will use to generalize (7, Theorem V.1, p 16]: |
Theorem 1 There exists a vector space V: O1G-D) CVC 2G’ — D) such that
Resply: V > IF} is an isomorphism
Proof Since Resplag_ p) is injective, it suffices to prove that Respla(œ~p) 1s surjective We have ker (Resplae-p)) = {n € O(G' — D): Resp,n = 0, v= 1 n} = AG’) Therefore ll rank (Resplacr—p)) dim 2(G’ — D) — dim 2(G’) g — 1 —deg(G’ — D) — (g — 1 — degG’) = degD=n,
using the Riemann-Roch-Theorem and the fact that 0 < dim L(G’ ~D) < dim L(G’) = 0
The problem of decoding is: For an arbitrary given y € FY, find c € C with minimal Hamming distance wt(y —c) According to Theorem 1, Resp gives a correspondence of vector spaces:
Q(G-D) c V c AG'-D)
[Reso [Reso
Cc c FF
If Fi ~ V, w + m, denotes the inverse map of Resply, then we can describe the
decoding problem in terms of differentials: For a given Ny, find n, € A(G — D), such
that Ne := Ny — N- has minimal number of Poles in P,, ,P,, ie such that ne may be written as a fraction of functions with low degree We will precise that in the next
section
Trang 14
2.3 The key equation
Let y = cte € FF} with € C and F an arbitrary divisor on X Let D, denote the unique
Divisor with 0 < D, < D and P, € supp D, © e, # 0, that is, D = ((ne)oo)luppd-
Definition 1 By a solution of the key equation we will denote any tripel (B,w,a) €
(E(#)\ 0) x 9(Œ' — F) x 0(G - D— FP), satisfying Bn, = a+ w
Proposition 1 7ƒ deg F + wt(e) < d*, then any solution (B,w,a) of the key equation
satisfies ne = § and He = 5
Proof Let (B,w,a) be a solution of the key equation From w = Bn, — a and n, = Note one concludes w — By, = By, — «a If the two sides of this equation do not vanish, we may estimate their divisors:
(w — By.) = min((w),(B) + (n-)) > min(G’ — P,G’ - F- D,.) = G'- F-D, (Bn ~ a) > min((B) + (n.),(a)) > min(-F+G-D,G-D-—F)=G-D-—F
Since D and G’ have disjoint supports and G < G", we get
(œ — Bne) > max(G’ - F- D,,G- D~ F)=G-—D,—F,
but deg(G~—D, — F) = deg G—wt(e)—degF > deg G—d* = 2g -2, what contradicts the assumption that w—Bn, is a non-vanishing differential Thereforew~Bn = Bn,—a = 0, what proves the statement
Proposition 2 If deg F > wt(e) +9, then there exists a solution of the key equation Proof By the Riemann-Roch-Theorem, dim L(F — D.) > 1+ deg F — deg D ~ g >
1+ wt(e) +g — wt(e) — g = 1, which guarantees the existence of B € L(F — D.)\ {0} c L(F) \ {0} Now we have Bn, = Bn + Bn, (Bn) = (B) +(n-) > —F + G — D, and (Bn, — Bn.) = (Bne) = (B)+ (ne) = ~-F+D.+G'—D, = G'— F, therefore (B, Bn., Bye)
is a solution of the key equation
Corollary 1 Let F be a divisor of degree |##1 For any y € IF) such that there is cCC with wile :=y—e) < |“, there exists a solution of the key equation On the other hand, for any solution (B,a,w) the equality n = $ holds
Proof One easily checks, that we have deg F +wt(e) < d* and deg F > wt(e)+g Now apply Propositions 1 and 2
The proofs of the Propositions show, that already the assumptions dim 0(G ~ D, —
F) = 0 resp dim L(F — D,) > 0 suffice to guarantee existence and uniqueness of solu- tions This coincides exactly with the assumptions needed by the well-known algorithm
(see [3])
2.4 Solving the key equation
Assume to := || to be non-negative (i.e d* > g) and F to be a divisor of degree |##=| = to +g For an arbitrary € IF? consider the linear map
6, : LiF) -— O(G'-D- F),
Bow B-ty
Notice, that A(G - D~ F) nN Q(G' — F) = Q(G — F) = {0}, since deg(G — F) =
deg G — deg F > deg G — d* = 2g — 2 Hence there exists a vector space W such that
Q(GŒ' — D - F) = 0(G - D - F)®@0(G' ~ P) @W (1)
Let tw, ™aq—r) denote the natural projections onto W resp 2(G’ — F)
If there is any codeword c with wt(y—c) < to, then, by the Corollary, there will exist
Be L(F)\ {0}, a € (1G — D— F) and w € 2(G’ — F) such that 6,(B) = Bn, = a+w
Every such triple will suffice , = % In this case, the following algorithm will compute the error vector e:
1 Compute the matrix describing 6,
2 Determine B € ker(xw o ế„,) \ {0} 3 Compute w := ta q-ry(6,(B))
4 Compute e:= Resp §
If too many errors have occured, then either ker(ry 045,) = {0} or the computed e won’t
suffice the conditions wt(e) < t and — e€ Ở
2.5 Realisation and complexity
Let (€,) <1 denote the canonical basis of IFy If the matrices describing 6, are computed before the algorithm starts and once forever, the matrix of 6, = 37", yde, may be computed with complexity order O(n*) in run time Then parts 2 and 3 of our algorithm may be done by simple linear algebra operations; the first one with
complexity order O(n*) and the second with order O(n?) We will now describe shortly
how to realize the remaining third part in O(n?) steps:
Trang 15e Compute B;, Bjis, until B;, 4 0 occurs
@ Compute aj,-1
At) 1
e Set Respỹ := ——- t0
Since w and B are given as linear combinations of certain basis differentials (resp
functions), any coefficient of their power series may be computed as an corresponding
linear combination if the power series at every point of supp D of any basis differential (resp function) is known a priori The complexity to compute one coefficient that way
is O(n) Altogether, there have to be computed
va +1+ p(B) +vp(F)) < 2n+ deg ((B) + F)huppp < 2n + deg F v=l1
coefficients, hence one can get along with complexity order O(n) in the decoding algo- rithm’s fourth part Furthermore, an analogous computation shows that not more than deg F coefficients of each power series around each point of the basis differentials (resp functions) must be known a priori
3 Essentially that’s nothing new
To demonstrate how our method of decoding is connected to that of [3] resp [4], we will
show that the main steps of each of the decoding procedures are equivalent We first consider the map
6:0(G' —D — F) + L(G — F) = {linear ¢: L(G — F) > F,},
where ®(7)(f) = Dv Resp, (f - 7)
Proposition 3 If the supports of F and D are disjoint, then Oly is an isornorphism.` Proof Since equation (1) holds, it suffices to show that
1 & is surjective,
2 O(G’ — F)@Q(G — D— F) C ker and
3 dimW = dim L(G — F)’
1.: The map L(G — F) > F?,f (ƒ(P\), , ƒ(P,)) has kernel L(G — D — F)=
{0}, hence is injective, therefore it suffices to prove that $:2(G’ —- D- F) ¬ (Fp), (n)(v) = x Resp, v,, is surjective Let (e”),=1., denote the canonical basis
of (IF})’ For any v, 2(G' — F — P,)\ Q(G' — F) is not empty as a consequence
of the Riemann-Roch-Theorem, and the image by & of any such differential is ey Hence @ is surjective, hence ©, too 1Proposition 3 together with decomposition (1) show the exactness of 0+ 2(G — D~ F)@Q(G’ - F) 4 AG’ - D- F) 4, L(G - FY’ +0, that may be also derived in a canonical way of a short exact sequence of sheaves on the curve X For details see [13]
2.: If y € Q(G’ — F) then for any v, Resp.n = , vĩ = 0, hence ®(n) = 0 Now take 7 € UG — D—#') and ƒ € L(G — F) Then fn € O(—D) and, by the residue theorem 1s: 3) Resp,(ƒn) = 0 dimW = dim0(G'— D~ F) ~ dim0(G — E) — dim 0(G — D — F) =_ g—1~deg(G'— D~ Ƒ) — (g— 1— deg(Œ' — F)) — —(g—1~ deg(G - D— EF)) : 1—g+ deg(G — F) dim L(G — F) = dimL(G~- FY), lI by the Riemann-Roch-Theorem, usin , g the fact that deg(G — F _ deg(G — D — F) < 0 at ) > 2g ~ 2 and
In our decoding procedure, the fundamental step is to find an element of ker(xự oô„)\
{0} Ww hi ch 1 Ss ? by P T OposI tì on 3 eq ul valent to findi ng somethi ng contai ned in k , er $ Ww Ọ Ww y) \ { 0 }- B ut | (Sw o mw 0 6,(B)) (f) (6,(B))(f) = ®(n - B)(/) >> Resp, (ƒ - nụ - B) =_ 3 )ƒ(P,)- B(P,) - Resp,(n,) = 3 ›ƒ(ŒF,)- BŒ,) - w,
for any B € L(F) and any ƒ € L(G — F) Hence Sly o mw o 6, is exactly the error locating map Ey in [4] Findi củ
v - Finding a non-trivial element of its kernel is t i :
the known algorithm, too is the main step in
4 Porter’s decoding algorithm
We will give a very short overview over Porter’s algorithm and explain why we consider the presented algorithm as its generalisation Porter makes some restrictions to the codes he treats We will formulate them using the notations of Section 2:
1, G' = —P,,, where P,, is a rational point on X, distinct from supp D 2 G is lineary equivalent to (deg G)- Pg, G+ Pop is effective and deg G > 0
3 (2g — 2)P is canonical
Especially assumption 3 won’t be realizable for most curves, but fortunately this re- striction may be dismissed easily
First remember the isomorphism Resp: V 7% i | tals fee p esp:V — IF of Section 3, and let €1, ,€, a
Trang 16let be a rational function? with (7) = G — (deg G).- Poo, wo a differential with (wo) =
(2g — 2)Poo, and m := —vp,,(7) For any “received” vector y = c+e € F}, Porter defines a rational function called “syndrom” by
s:=¥ 1- 5) ¬ Š{ WP.) ) to
Note that S € R := Uso L(tP.), Swo = ny (mod 7), ie Swo = ny + vf for some
f € R, and —vp,(S) < m+ 2g — 1
In order to decode Porter looks for solutions of the “polynomial congruence” A — BS =0 (mod 7) subject to a certain “condition at P,.” More precisely, one may
describe this by*®
If there is an integer t <d*~ wt(e) and A, B, C € R satisfying —vp,,(A) <
t+2g—1 and —vp,(B) <t such that
A—BS=CŒ+, (2)
- then je = fwo
In the following, we give a short description of Porter’s method of finding such A, B,C Consider the linear map
: L(tPao) @L((t-+ 2g —1)Poo) > L(m+t+2g ~1)P.») (B,C) - BS + Cy
The solutions of (2) clearly will satisfy ~vp,,(C) < t+ 2g —1, therefore they coincide
exactly with the tripels
' §(B,C),B,C where 6(B,C) € L(t+2g—1)
If one chooses elements ¢; € R such that
L{TP ) = span {d1, " ›Ôaim1(xPe)} for any 7 €IN,
then the matrix M, describing 6 with respect to those bases, is the transpose of Porter’s resultant matrix Clearly, searching for a solution of (2), may be done by looking for a non-trivial linear combinition of the columns of M with zeroes in the m lower positions One way to do that is Porter’s “row reduction process” at M7, but clearly there are several other methods
We conclude by explaining, how Porter’s method and the one described in section 2, are connected Multiplying equation (2) by wo and substituting Awo by w, Swo by go and Cw by a, one do not need neither the differential wo nor assumption 3 for the equivalent statement
2In genus 0 case, y is the Goppa-Polynomial, if P,, denotes the infinite point of the projective line - #This are sligthly weaker assumptions than those in the original, where only solutions of minimal degree (= —vp,,) are taken into account; for more detailed information see [13]
If there is an integer t < d* — wt(e) and B € L(tP.), w € Q(-(t + 1) Peo) such that w ~ Bo = ay for some a € (X), then ne = §
The underlying idea of this is to write 0 = % (mod +) where B has few zeros, i.e few poles More directly and without the need of S* one may, provided a given 7,, search
for w and B such that 7, = $ (mod 2(G— D)) and B has few poles (e.g (B)+F 2 0
for some “small” divisor F) This is the main idea of the method of decoding described
in Section 2
References
(1] J.H van Lint, G van der Geer, Introduction to Coding Theory and Algebraic
Geometry DMV Seminar, Band 12 Birkhauser Verlag 1988
[2] J Justesen, K.J Larsen, A Havemose, H.E Jensen, T Hgholt, Construction and
decoding of a class of algebraic geometry codes IEEE-IT 35(4)(1989), pp 811-821 "BỊ A.N Skorobogatov, $.G Vladut, On the decoding of algebraic-geometric codes IEEE-IT 36(5)(1990),pp 1051-1060 [4] R Pellikaan, On a decoding Algorithm for Codes on maximal curves IEEE-IT 35(6)(1989), pp 1228-1232 [5] S.G Vladut, On the decoding of algebraic-geometric codes for q > 16 IEEE-IT 36(6)(1990), pp 1961-1963
{6] S.C Porter, Decoding Codes arising from Goppa! S construction on algebraic curves Thesis, Yale University, 1988
[7] S.C Porter, Decoding Geometric Goppa Codes Preprint
[8] S.C Porter, Euclid’s algorithm, resultants and rational function representation on algebraic curves with a single point at infinity Preprint
[9] S.C Porter, Dense representation of affine coordinate rings of curves with one point at infinity Proceedings of ISSAC-89
{10] S.C Porter, An efficient data structure for rational function on algebraic curves
Preprint
[11] S.C Porter, B.Z Shen, R Pellikaan, Decoding geometric Goppa codes using an extra place, Preprint Eindhoven University, September 1991
[12] B.Z Shen, Subresultant sequence on a Weierstrass algebra and its application to decoding algebraic-geometric codes preprint Eindhoven University, May 1991 [13] D Ehrhard, Uber das Dekodieren Algebraisch-Geometrischer Codes Thesis,
Dũsseldorf University, 1991
Trang 17
On the Different of Abelian Extensions of Global Fields
G Frey, M Perret, H Stichtenoth O Introduction
Let q be a power of some prime number p, and let F, be the field with g elements
Coding theorists are interested in explicitly described function fields over F, having
a large number of F,-rational places (or, equivalently, irreducible complete smooth algebraic curves over F, with many F,-rational points) For small values of the genus,
such function fields are often abelian extensions of the rational function field F,(z) For
instance, this is the case for Hermitian curves, some Fermat curves, and some Artin-
Schreier extensions of F,(z) Moreover, one way to exhibit families of function fields
E/¥, of genus growing to infinity and having good asymptotic behaviour (i.e., the ratio
(number of rational places/genus) has a limit > 0), is to construct a tower of function
fields Ey C Fy C Ey over F,, each step E;4,/E; being Galois with an abelian Galois group In other words, solvable extensions may have a good asymptotic behaviour, cf [3]
One aim of our paper is to show that abelian extensions E; /F (where F is some fixed function field over Fy, and F, is assumed to be the full constant field of F and all địt > 1) are asymptotically bad (i.e., the ratio (number of rational places/genus) tends _ to 0 as the genus of E;/F, goes to infinity)
It should be pointed out that our method uses only elementary results from Hilbert’s ramification theory, cf [2,4], and the finiteness of the residue class fields In the case of global fields, one may also use class field theory in order to obtain some results of this paper
1 Hilbert’s Ramification Theory for Locally Abelian Extensions
In this section, we consider the following situation K is some field, 0 C K a discrete
valuation ring and g C o the maximal ideal of o Let L/K be a finite abelian field extension with Galois group G (i.e L/K is Galois, and its Galois group G is abelian)
Let O C L be a discrete valuation ring of L with o C © and maximal ideal P, hence 9 = PNo Throughout section 1, we suppose that © is the only discrete valuation ring
of L containing o Let k := 0/p and 1:= O/P denote the residue class fields of o resp O Then I/k is a finite field extension, and we shall always assume that I/k is separable We choose some P-prime element 7 € P (i.e., P is the principal ideal generated by 7),
and consider the groups
Go:={ơ€G|øz=z mod P forall cE 0}
and, for i > 1,
G¡:={ơ€Ga|lơr=mz mod Pit},
It is well-known that the definition of G; is independent of the choice of 7, and G2DGo) DG D D Ga = {1} for sufficiently large n > 1, see [2,4] The factor groups mi/ptt (for i > 0) are considered as vector spaces over | via
(c+ P)-(a+ Pit) = ca + PH (2 € O,ae P’),
and G acts on P'/P't! by
r(a+ Pt) := (a) + PIt?
(in order to see that this action is well-defined observe that O is the only extension of
o in L, hence r(P) = P for all r € G) We set
Xi := {a+ Pt) € p/p} | r(a+P!) =a+P* forall 7 € G} Clearly, X; is a k-subspace of P*/P**
Proposition 1: The dimension of X; as a vector space over k is at most one
Proof: By Hilbert’s ramification theory I/k is a normal field extension Due to our
assumption I/k being separable we obtain that I/k is Galois Moreover, any automor-
phism 79 in the Galois group of I/k is induced by some r € G, i.e To(z +?) =r(z)+?P
for any z +? € O/P = l, see BỊ In order to prove the proposition we can assume
that X; # {0} We choose a + Pit! € X; with a € P'\P*t) Since P P is a
one-dimensional vector space over I (this is obvious) we have for all a, + Ppt EX:
a, + Pit! = (c+ P)- (a+ Pt) for some c € O For any 7 € G, the following holds:
(c+P)-(a + Pitt) =a,+ pit — r(a, + Pitty
=1(e+P)-1(a+ Pit) = r(c+ P)- (a+ Pit),
consequently r(¢ + P) =c +P for any r € G Thus c+ P is invariant under any auto- morphism of I/k, i.e c+ P € k This proves Proposition 1 "
It is well known that the mappings Ủ: { of GQ—
o —» mod P
resp for? > 1
pi: 19m —: 7!/P*\ ơ — ##—1 mod Pit
Trang 18Proposition 2: Under the hypotheses of this section, the image of is contained in k*, and the image of y; is contained in X; for any i > 1
Proof: (a) Let o € G We have to show that 7o(()) = (Ø) for all 7» in the Galois
group of I/k As before, 7 is induced by some r € G, and we obtain
(we have used that G is abelian and r(z) is a P-prime element as well)
(b) An analogous argument proves that y;(o) € X; for any o € G; " We recall some facts from ramification theory Let f = f(P | ø) = [I : k] denote the residue class degree and e := e(P | p) the ramification indez of P over 9, 1.e pO = Pe,
Since O is the only extension of o in L and I/k is separable, we have e- f = [Z : K] Let s:= char(k) be the characteristic of the residue class field and gi = ord G; for any t > 0 Then G; is the unique s-Sylow subgroup of Go, and go = e The extension P lp
is said to be tame if g, = 1 (hence (s,e) = 1), otherwise P | p is wildly ramified
Let W C k* be the group of roots of unity in k If W is finite, we set w := M7 Corollary 3: In addition to the hypotheses of this section, suppose that k contains only finitely many roots of unity If P | pis tame then e < w
Proof: Consider the map : Go — I* as before Since G, = 1, is a monomorphism By Proposition 2, the image of » is contained in W In order to obtain a similar estimate for the ramification index also in the case of wild ramification, we introduce the following notion: an integer i > 0 is called a jump (for
P |) if Gi # Gigs
Corollary 4: In addition to the hypotheses of this section, assume that k is a finite
field Then e < (#k)" where r denotes the number of jumps ,
Proof: e = go = (90/91) -(g1/92)- - (9n/9n41) where n is chosen such that Gnt1 = 1
Proposition 1 and 2 yield g;/gi41 < #k for any i > 0 The corollary follows immediately.s
Hilbert’s formula [2,4] states that the different exponent d := d(P | 9) is given by
d= Ð (ø¡ — 1)
t>0
This formula can be restated as follows We consider the set {Ha, ,vr} of jumps
(where 0 < < ¿ <: < ưẹ and r is the number of jumps) and set hi=m+l f:=U¡—Ui¿-g for t=2, ,7 Then d=) > ti(gy, — 1) =1 Since G is abelian, the Hasse-Arf theorem [2] applies It yields f-g„, =0 mod e
for i= 1, ,7 Combining this with Hilbert’s formula we obtain
Proposition 5: Under the hypotheses of this section, the different exponent d satisfies the estimate d> sre where r denotes the number of jumps ` Proof: r d=) tigy,(1 — 97,3) i=1 1< 1 25° 2 ta 2 re
by the Hasse-Arf theorem "
2 The Different of Abelian Extensions of Global Fields
In this section, F denotes a global field This means that either F is a number field, or
F is an algebraic function field of one variable over a finite field F, (we assume that F, is the full constant field of F) A place of F is the maximal ideal of a discrete valuation ring of F If g is a place of F, its corresponding valuation ring will be denoted by 0p The residue class field 0,/p is a finite field, and in the function field case we have
Fy € o, The degree of 9 is defined by
deg p := log #(0,/p)
(in the number field case, log is taken with respect to the basis e = 2,718 ; if F isa
function field over F,, we take log = log, - the logarithm with respect to the basis q) The definition of the degree is extended to divisors of F (a divisor is a formal finite sum
of places) by linearity
Trang 19Let E/F be an abelian extension of F and Gal(E/F) be its Galois group If ø is a place of F, there are g = g(g) places Pi, ,P, of E lying over g (i.e C P,) All of them have the same ramification index e(g) := e(P, | g) and the same residue class degree ƒ(p) := f(Pv | p), and we have e(p) - f(y) - 9(g) = [E: F] For an extension P = P,
of pin E/F, we consider the decomposition group
G(ø) := G(P | p) := {o € Gal(E/F) | oP = P}
(this is independent of the choice of the extension P since E/F is abelian), and the decomposition field Z = Z(p), ie FC ZC E and G(p) = Gal(E/2)
There existis the unique rnaztmal tunramificd subeztension F CMCE This means that all places of F are unramified in M/F, and M is a maximal subfield of E with this
property Let S := S(E/F) be the set of places of F which are ramified in E/F (it is well-known that S' is finite)
Lemma 6: With the notations as above, we have
» log e(p) > log[E : F] — log[M : F]
pes
Proof: For any g € S, let Go(g) C G be the inertia group of 9, see [2,4] Its order is e(p), and if U C G is the subgroup of G generated by all Go(g) (with p € S), then M is the fixed field of U Therefore ord U = [E: M] = [E: F]/[M : F] Since G is abelian,
ord US [Tord G(p) = J] e(p)
pes pes
Taking logarithms yields the assertion of the lemma " Let D(E/F) be the different of E/F The main result of this section is the following: Theorem 7: Suppose that E/F is an abelian extension of global fields and
F CM C E is the maximal unramified subextension In the function field case we assuine, in addition, that E and F have the same constant field F, Then the degree
of the different D(E/F) satisfies
deg D(E/F) > 52 : F]- (log[# : F] —log[M : F})
Proof: For p € S, let d(g) be the different exponent of a place P of E lying over g, and r(g) be the number of jumps, cf section 1 (observe that we can apply the results of section 1 if F is replaced by the decomposition field Z (g)) We obtain
deg D(E/F) = 3” Ð d(p) - degP PES Plp = À_ø(ø) - d(9)- f(p) - degp pes > 2 3 66) -?(ø) -e(8)- ƒ(ø) -degp —_ (by Proposition 5) pes Re dle Nile {E: F]- >> r(p) - deg p pes [E: F]- 3> !ose(p) pcs [E: F] - (og[E : F] — log[M : F)) (by Lemma 6) ; > (by Corollary 4) >> ~9
3 Abelian Extensions of Function Fields Are Asymptotically Bad
We want to prove a slightly more general result than we announced in the introduction
For an algebraic function field E/F, (with F, as its full constant field) we set
g(£) = genusof E
N(£) = number of rational places of E/Fy
If E/F is a Galois extension with Galois group G, we let G' be the commutator subgroup of G The fixed field E* D F of G' is the mazimal abelian extension of F contained in E In particular, if G is abelian, G' = {1} and E® = E
Theorem 8: Let F/F, be an algebraic function field and (E,),>1 be a sequence of
extension fields of F' with the following properties:
(i) Fy is the full constant field of F and all E, (i) E,/F is Galois with Galois group G, (iii) ord (G,/G,) — œ as w —> o0
Then the quotient N(E,)/g(Z,) tends to zero as vy —+ 00
Proof: There is a constant h (the class number of F) such that any abelian unramified extension M/F with the same constant field Fy is of degree |M : # <A, cf {1} We
consider the maximal abelian extension F, C E, of F contained in E, By (iii), the
Trang 20dy > sny(logny — log h)
by Theorem 7 The Hurwitz genus formula yields
a(Fe) 2 m(g(F) ~1) + 24,
> mu(g(F) —1+ 2 (ogn, log h)
Ôn the other hand, we have the trivial estimate N(Fy) < nụ - N (F), hence
NUP) _ N(F) oo
9Œ) — g(F)— 1+ } (log nụ — log h) for —+ oœo Eventually, since N(E„) < [E, : Fy] - N (F,) and
9Œ) 3 [E, : FyÌ(@(F,) — 1) > }[E, : F,] - g(Fy) (observe that øŒ,) —¬ 00 for
-# — 00), we obtain N(E,)/9(E,) — 0
References
{1] Artin, BE and Tate, J.: Class field theory New York - Amsterdam 1967 [2] Serre, J.P.: Corps locaux Paris 1962
[3] Serre, J.P.: Sur le nombre des points rationnels d’une courbe algébrique
sur un corps fini C.R Acad.Sc Paris, t 296 (1983), 397-402
[4] Zariski, O and Samuel P.: Commutative Algebra, Vol I Princeton 1958 Gerhard Frey Institut fiir Experimentelle Mathematik Universitat GHS Essen Ellernstr 29, D-4300 Essen 12 Germany Marc Perret
Equipe Arithmétique et Théorie de l’Information CIRM, Luminy Case 916 F-13288 Marseille Cedex 9 France Henning Stichtenoth Fachbereich 6, Universitat GHS Essen D-4300 Essen 1 Germany
Goppa Codes and Weierstrass Gaps
Arnaldo Garcia R.F Lax
IMPA Department of Mathematics
Estrada Dona Castorina 110 LSU
22.460 Rio de Janeiro Baton Rouge, LA 70803
Brasil USA
We generalize an example of Goppa to show how the gap sequence at a point may often be used to define Goppa codes that have minimum distance greater than the usual lower bound
1 Let X denote a nonsingular, geometrically irreducible, projective curve of genus g
defined over the finite field F, with g elements Assume that X has F,-rational points Let D be a divisor on X defined over F, (i-e., D is invariant under Gal(F,/F,)) Then E(D) will denote the F,-vector space of all rational functions f on X, defined over Fy, with divisor (f) > —D, together with the zero function, and Q(D) will denote the F,-
vector space of all rational differentials 7 on X, defined over F,, with divisor (ny) > D,
together with the zero differential Put 1(D) = dimy,L(D) and i(D) = ding,Ô(D)
The Riemann-Roch Theorem states that
\(D) = deg D+1—g+4(D)
Also, we will write Do (resp Do.) for the divisor of zeros (resp divisor of poles) of D Hence we have
Do > 0, Da 2 0,(Supp Do) N (Supp Doo) =, and D = Do — Doo
V D Goppa [3,4] realized that one could use divisor theory on a curve to define nice codes A q-ary linear code of length n and dimension & is a vector subspace of dimension k of F? The minimum distance of a code is the minimum number of places in which two distinct codewords differ For a linear code, the minimum distance is also the minimum weight of a nonzero codeword, where the weight of a codeword is the number of nonzero places in that codeword A linear code of length n, dimension k and
minimum distance d is called an [n,k, d]-code Let C denote an [n, k, d]-code over Fy
A generator matrix of C is a k x n matrix whose rows form a basis for the code A parity check matrix for C' is an (n — k) x n matrix B of rank n — k such that AB* = 0 for some generator matrix A of C Two codes C and C* are called dual if a generator matrix of C* is a parity check matrix of C The code C has minimum distance d if and only if in any parity check matrix for C every d ~ 1 columns are linearly independent
and some d columns are linearly dependent '
The first author was supported by the Alexander von Humboldt-Stiftung while visiting Universitat GHS Essen within the GMD-CNPq exchange program The sec-
ond author was partially supported by a grant from the Louisiana Education Quality
Trang 21s Let G be a divisor on X defined over F, and let D = Py + Py + -+P, be another divisor on X where P;, ,P, are distinct F,-rational points and none of the P, is in
the support of G The geometric Goppa code C(D, G) (cf [6]) is the image of the linear map a: L(G) —+ F} defined by
fro (f(P1), f(P2), -, f(Pa))-
The geometric Goppa code C*(D,G) (cf [6]) is the image of the linear map a* : 0(G — D) - F} defined by
7 +— (resp, (n), resp, (7), , resp, (7))
The codes C(D,G) and C*(D, G) are dual codes ((6, 11.3.3]) Alternatively, if one fixes a nonzero rational differential w with (canonical) divisor K, then, the image of a* is
the same as the image of the map Ø* : L(K + D ~ G) — F, defined by
ƒ (resp, (fw), resp, (fw), , resp, (fw))
"We will be mainly interested in the code C*(D,G) This code has length n and dimension
k= dim L(K + D —G) — dim L(K —G)
In particular, if G has degree greater than 2g — 2, then &k = dim L(K+D-G) Ifk > 0,
then the minimum distance d of C*(D, G) satisfies the well-known bound d > deg(G) — 2g +2
2 Now let P denote a (closed) F,-rational point on X and let B be a divisor defined over F, We call a natural number + a B-gap at P if there is no rational function f on X such that ((f) + B), = 7P We say that + — 1 is an order at P for the divisor K — B (where K denotes a canonical divisor on X) if we have K-B~(y-1)P+B,
where E > 0, P ¢ Supp(E), and ~ denotes linear equivalence It follows from the
Riemann-Roch Theorem that is a B-gap at P if and only if + — 1 is an order at P for the divisor K — B Note that with this terminology, the usual Weierstrass gaps at P are the 0-gaps at P
Let WG and 7, be B-gaps at P Put GŒ = (ị +13 — 1P +2B (1) Our divisor D used to define the code C*(D,G) will be of the form D=h;+PhB+ -+P,, where the P; are n distinct F,-rational points, each not belonging to the support of G
(2.1) THEOREM Assume that the dimension of C*(D, G) is positive (where G is of the form specified in (1)) Then the minimum distance of C*(D,G) is at least deg(G) — 2g + 3
PROOF Put w= deg G— (2g — 2) If there exists a codeword of weight w, then there exists 7 € 2(G — D) with exactly w simple poles P;, P2, , Py in Supp(D) We then
have
(n)o > Go and (n)« < Ge + Pì + P + + Pu
Hence, 29-2 = deg(n) > deg Gp — deg Goo — 9o = 2g — 2 It follows that (j)o = Go and
(n)se = Ges + Pị + Pạ + - + Pu Thus there exists a canonical divisor K of the form K=G-(h+P›+ -+ Pu) (2 Since + is a B-gap at P, we have | K-B~(%-1)P+E, (3) where E > 0 and P ¢ Supp(£) From (1), (2), and (3), we have E+(PitPat-:-+ Po) -(B+7P) ~ 0
Thus there exists a rational function f such that ((f) + B)o = 7;P, contradicting the
fact that y; is a B-gap at P 4
(2.2) REMARK If B = 0, then degG = 7; + — 1 and it is natural to ask which
integers can be written as the sum of two Weierstrass gaps In this direction, G Oliveira [7] showed that if X is nonhyperelliptic, then each r, for 2 < r < 2g, is the sum of two gaps, with the exception of r = 2g ~— 1 in the case that 7 = 2ø — 1
Let @: X — PÄ be a morphism with nondegenerate image As in [10], we may
view @: X > P* as a parametrized curve in P* and the points of X.as its branches To any point P € X, one associates a sequence of natural numbers
€9(P) < 41(P) <-+- < en(P),
which are the possible intersection multiplicities of the curve y(X) with the hyperplanes in P® at the branch centered at P We denote by rÿ"x (cf [10]) the osculating
space of dimension j ~ 1 at P (associated to the morphism ¢); thus,
Lÿ~”X = (Ì(H : H is a hyperplane with I(P; o(X) - H) > ¢;(P)},
where I(P; y(X) - H) denotes intersection number
Assume from now on that the linear system |K — B| is base-point-free If 71(P) < %a(P) < - < Tw++(P) are the B-gaps at P, then for the morphism associated to the linear system |K — B|, we have
Trang 22(2.3) THEOREM With notation as in Theorem (2.1), assume moreover that GH =
7; +1 and that the osculating space Lg-Yx (with respect to the morphism defined by
|K — B|) does not meet the curve p(X) at another F,-rational point If C*(D, G) has
positive dimension, then its minimum distance is at least deg G — 2g + 4
PROOF By Theorem (2.1), the minimum distance of C*(D, G) is at least deg G—29+3
Put w = deg G — 29 + 3 and suppose that there is a codeword of weight w Proceeding
as in the proof of Theorem 1, we see that there is a canonical divisor K of the form K=G+Q-(Pit Prt +-+Pu), | (4) where Q is an F,-rational point on X Using equations (1), (3), and (4), we get B~-*~;P- Q+(hị tị +‹ + Pu) + E Since 7; and 7;+1 are both B-gaps at P, we see that Q ¢ {P, Pi, Pe, ,Pw}U Supp(£) Hence, LB +7;P) #4 L(B+74P+Q) By Riemann-Roch, we then have }(K ~ B~ +¡P) = L(K - B - +¡P - Q)
This equality means that for any divisor A linearly equivalent to K — B, if A > WP, then A > 7;P+Q Now, s;(P) = 1+1 —l =3, soif A > €;(P)P, then A > s;(P)P+q It follows from this that Q is in Lg) x , contradicting our hypothesis g
(2.4) REMARK Suppose B = 0; i.e., yg : X —+ P9-! is the canonical morphism If at
an F,-rational point P, we have g = 2g — 1, then Lg-)x , the osculating hyperplane at P, meets X only at P and hence rÿ-Đx, for 7 = 1,2, ,g — 1, meets X only at
P, so the condition on the osculating space in Theorem (2.3) will be satisfied The next
result gives another useful criterion for verifying this osculating space condition
(2.5) PROPOSITION Suppose that B = 0 and that X is not hyperelliptic For
j € {1,2, ,9 — 2}, we have
1 = 2j = (LỆ~”X)n X = {P)
PROOF First we remark that 734; < 2j for j = 1,2, ,g—2 For suppose 7;41 > 2j
Then there are at most j gaps less than or equal to 2j Hence the dimension of L(2jP) would be at least 27 + 1 — j = j + 1, contradicting Clifford’s Theorem
Now, the intersection divisor of X and rÿ-)x is of the form A = (1;+ —1)P+ E, where E > 0 and P ý Supp(E) By the geometric version of the Riemann-Roch
Theorem [1, p.12], we have
dim L(A) = deg A — dim LY~)X = deg A — (j — 1) (5)
From Clifford’s Theorem, we have
dim L(A) < (deg A)/2 +1 (6)
From (5) and (6), we get deg A < 2j —1, which implies that deg E is at most 2j —7;41
So, if 7341 = 2j, then F = 0 and the Proposition follows y
We note that Proposition (2.5) was proven by G Oliveira [7] in the case j = g — 2 (2.6) EXAMPLE We give an example to show that the converse of Proposition (2.5)
is false, and that one can sometimes deduce the fact that Lg-9) NX = {P} by testing
the condition 7141 = 21 for some I > j
Let X denote the nonsingular curve with function field Fg4(z, y) where y® +y = z° If P = (1,a), with a® +a = 1, is an Feq-rational point, then by Garcia-Viana [2] the
gap sequence at P is 1, 2, 3, 4, 5, 10, 11 Taking | = 5, we see that 74 = 10 = 21; hence, L@x meets X only at P It follows that Lx must also meet X only at P, but notice that for j = 4, we have 741 = 5 4 27 = 8
In the case where the next ¢ consecutive integers after 7; are also B-gaps, one can
prove results similar to Theorem (2.3) by assuming more conditions on the osculating
spaces -
(2.7) THEOREM Suppose that 7;42 = 7; +t for t = 1,2 Suppose that LDx does
not meet X in another F,-rational point Also suppose that Lÿ)x does not meet any of the following lines:
(i) A line joining two other F,-rational points
(ii) A line joining two F,2-rational points that are conjugate over Fy (iii) A tangent line at another F,-rational point
If C*(D, G) has positive dimension, then the minimum distance of C*(D, G) is at least
deg G — 2g +5
PROOF Put w = deg G — 2g +4 Since rÿx does not meet X at another Eq-rational
point, the same is true for rÿ)x - Thus by Theorem (2.3), the code C*(D,G) has minimum distance at least w Suppose that there is a codeword of weight w Then
there is a canonical divisor K of the form
K=G+A-(Pit+Pot +Py), | (7)
for some positive divisor A = Q; + Q2 of degree 2 defined over Fy From (1), (3), and (7), we have
B~ —yP—(Qi+Q2)+ (P+ Pet -+ Pu) t E, (8)
where > 0 and P £ Supp(E) : -
Now, either Q and Q are both F,-rational points or they are F2-rational points that are conjugate over Fy Since 7; +2 is a B-gap at P, we cannot have Q1 = Qo = P
By applying the proof of Theorem (2.3) to the divisor G+ P (note that 41 and y;41+1
are B-gaps), we see that neither Q; nor Q2 can equal P, since then the other point would
lie in Lÿ)x - Also, we have that neither Q¡ nor Q¿ is in {P\, Pạ, , P„}U Supp(#) It then follows from (8) that we have
Trang 23where these are now vector spaces of rational functions defined over Fy It follows from the Riemann-Roch Theorem that 1(K ~ B~ +ịP ~ Qu) = L(K - B ~ +;P — Qị — Q2) Thus for any divisor H ~ (K — B), we have H2>4jP+Q1=6(P)P+ Qi = H>y4P+Q14+Q2 =6(P)P + Qi + Qo Therefore, the osculating space LG-Yx meets the line joining Q, and Qo, contradicting our hypothesis §g
The conditions on Lg-Yx in Theorem (2.7) can be expressed as: Lg-) X misses all the lines determined by degree two F,-rational divisors A, where P ¢ Supp(A) By
using induction, one can generalize Theorem (2.7) in the case of t consecutive gaps after 7; to obtain the following result
(2.8) THEOREM Suppose that 7,42 = 7; +t for some natural number t Suppose that for each s € {0,1, ,¢—1} the osculating space Lg) x misses the linear space spanned by each degree t — s F,-rational divisor with support disjoint from {P} If C*(D, G) has positive dimension, then its minimum distance is at least deg G — (2g — 2) + (£+1)
3 We present some examples illustrating the theorems in the previous section:
(3.1) EXAMPLE Goppa [4, pp 139-145] considers the plane quintic y*z + yz* 4- z® — x?z5 over the field Fs This curve has 17 points over F, Put P = (0,1,0) Then
the gap sequence at P is shown to be 1,2,3,6,7,11 Goppa shows (p 144) that the code C*(D,16P), where D is the sum of the remaining 16 F4-rational points, has minimum
distance at least 8 This follows from Theorem (2.3) Our proofs of Theorems (2.1) and (2.3) are essentially generalizations of Goppa’s argument
(3.2) EXAMPLE Let X denote the curve y7 + y = xt defined over F,2, where
q 2 5 This is an example of a Hermitian curve There are gti F2-rational points
on X, the maximum possible (by the Hasse-Weil bound) on a smooth curve of genus 9 = (q —1)q/2 over Fy2 Let P be an F,2-rational point on X and let D denote the
sum of the remaining q* F2-rational points The codes C(D, sP) have been studied
extensively by Tiersma [11] and Stichtenoth [9], and recently Yang and Kumar [12] have
determined the exact minimum distances of these codes In this example, we show how to use Theorem (2.7) to give an alternate derivation of some of the results of Yang and Kumar
Let P = (a,b) be an F,2-rational point on X The canonical morphism y : X — P9~! given by a Hermite basis at P of the regular differentials (cf [2,8]) is y((x,y)) =
(1:(z—a): -: P*ˆ*: (e— a)4ˆ*: (z— a)? PI prs :(c—a)Ptr3; P+}), where P = (y—b)— q1(z - a) is the tangent line at P We note that P meets X only at P (with multiplicity q + 1) Denoting by P„, the point on X at infinity, we have that
9(Poo) = (0:0: -:0: 1) By [8] (or see [2]), the Weierstrass nongaps at P smaller
than 2g are given by: 0 qq+1 2q,2q + 1,2(q¢+1) (q — 4)q, (q — 4)q+ 1, , (g— 4)(g + 1) (q— 3)q, (qT— 3)q+ 1, , (q— 3)(q+ 1) (q— 2)q,(q— 2)q + 1, ,(g— 2)(g+1) -
Take j = g — 5, so that 7; = (q— 4)(q+ 1) +1 Note that 13:75 +1, and +; +2 are
three consecutive gaps :
Let Gy denote the divisor (7; +%—1)P = (%+ (q—4)(q¢+1))P and let D denote
the sum of the other g° F,2-rational points on X The usual lower bound on Goppa
codes would show that the minimum distance of C* (D, Gx) is at least deg G, —(2g—2) = Ye — 2(q+1) We claim that the conditions of Theorem (2.7) are satisfied here, so that the minimum distance of C*(D,G,) is at least 7, — 2(q +1) +3
To establish this claim, first note that Lg~) x NX = {P}, since y = 2g — 1 Next, we will see that Lg-) X= rp~®x (i.e., the osculating spaee of cođimension 5 at P) does not meet any line determined by two (distinct) F,-rational points (different
from P) Since ¢ is given by the Hermite basis associated to the point P, the osculating space Lự~®9x in P9-! is given by Xg-1 = Xy-2 = + + = g—s = O (where the X; are homogeneous coordinates) Suppose ~({t1,91) and (x2, y2) are distinct points on
y(X), both different from ¿(P) The line determined by these points is {ap(21,m1) + By(22, 92) : a, 6 € Fy}
Put _ |
P, = P(zj,y:) = (0i — b) — a*(%; — a) for ¡ = 1,2 If there is a point on the above line that also lies in L§-9x , then we have
œP‡ + 8P¿”° =0
(+) a(z, — a) Pe? + B(x — a)B£”Š = 0
œ£"? + 8£”? =0
Note that Pị # 0 and Py + 0; otherwise, there would be another point on the curve
(the point (z;, ;)) lying on the tangent line to the plane curve X at the point (a, 8)
Multiplying the first equation in (+) by Pi and comparing the resulting equation with the third equation in (*), we have P; = Pp, Multiplying the first equation in
(*) by (#¡ — a) and comparing the resulting equation with the second equation in (+),
Trang 24This contradicts the assumption that y(z1,y1) and y(z2, yz) were distinct points One argues similarly if one of the two points (2;, y;) is P>
Finally, suppose (),y;) is an F,,-rational point distinct from the point (a, 8) In order to determine the tangent line to p(X) at (21,41), we take derivatives with respect to z: d dP = (v1, 91) = zf and is (21,41) = (x1 — a)? Put P, = P(z1,y1) From the definition of ¿, the tangent line to y(X) at (21,41) is {ay(21,31)+ BV: 0,8 € F,}, where V =
O:1: -:(q—3)(a,—a)*PE* ; P23 1 i 4 (q—3)(z, —a) 41 Po i (q—2)(zì ~ a)1Pˆ~3) 1
Suppose there is a point Q on this tangent line that also lies in L's) x Since no rational point on X except (a, 6) lies in this osculating space, we see that 8# 0at Q, so we may assume ổ = 1 at Q Then at Q we must have
{or +(- (a — a P+ a(n — PI + PE? + (g— 3)(a — a) PF = = 0
Multiplying the first equation by (x; — a) and comparing the resulting equation with the second equation, we see that P, = 0 But this says that (71,41) belongs to the tangent line to X at (a,b), a contradiction Again, one can argue similarly if (x1, ¥1) is
Poo:
Since the projective change of coordinates (x : y : z) + (x: z: y) induces an automorphism of X that takes (0 : 1: 0) to (0: 0 : 1), we see that the conditions
of Theorem (2.7) are also satisfied at P Put d; equal to the minimum distance of
C*(D, Gy) If we take yy, = (2+m)q—1, where 1 <m < q—3, then by Theorem (2.7),
we have d, > mq Put z = (x—a,)(r—a2)-+- (%—am), where a1, @2, , đự, are distinct
elements in F,2 Then the differential đz/z has a zero of order (q?-q—-2)+mg > deg G,
at P and has exactly mq simple poles, so it is a codeword of weight mq Thus, the
minimum distance of C*(D,G;) will be exactly mq when Ye = (2+m)q—1 Using
Riemann-Roch to compute the dimension of this code, we see that C*(D,G;) is a
ÍqẺ,gẺ +4 — q(q+ 2m — 1)/2, mq]-code when + = (2 + rn)g — 1
4 Although our theorems show that the minimum distance of the code C*(D, sP) is
related to the Weierstrass gap sequence at P, we close with an example to demonstrate that this minimum distance does not depend solely on the gaps Specifically, we give a curve with two points with the same Weierstrass gaps such that corresponding Goppa
codes have different minimum distances
(4.1) EXAMPLE Let X denote the nonsingular plane quartic af + yz 42727 + rzi + 2zy2z = 0 over the field F; Then X has the following eight F5-rational points: P= (0, 1,0), P, = (3,1,1), P3 = (4,1, 1) y= (3, 2, 1), P= (4, 3, 1), Tạ = (2,4,1) P =(0,0,1), and P’ = (2,3,1)
The points P and P’ are (ordinary) flexes of X, so the Weierstrass gap sequence at each
of these two points is 1,2,4 Put D = Py+P2+ -+Pe+P! and D! = Py4+Po+ -+Ps+P The functions 1 and z/z are a basis for L(3P) and a generator matrix for the code C(D, 3P) is 1 1 1 1 1 '1 1 0 2 4 2 4 3 3/ˆ It can then be seen that the weight enumerator (cf [5, p 38]) of C(D,3P) is 1+ 122° + 42° + 827 By the MacWilliams identity (cf [5, p 39]), the weight enumerator of the dual code C*(D,3P) is 1 + 122? + 8029 + 400z* + 8042° + 1180z° + 64827 The functions 1 and (x + z)/(y + 2z) form a basis of L(3P’) A generator matrix for C(D',3P’) is then i1 1 1 1 1 1 1 0 3 010 3 3/ˆ The weight enumerator of C(D’,3P’) is then 1+482* +426 4 1227 Thus C(D,3P) and C(D’,3P") have different minimum distances However, the weight enumerator of C*(D',3P’) is ‘ 1+ 24z? + 6025 + 36024 + 924z° + 10802° + 6762’,
and so C*(D',3P') has the same minimum distance as C*(D,3P)
Trang 25The functions 1, (z + z)/(w + 2z), and (£ + z)( + 3z)/(w + 2z)? form a basis for L(5P") A generator matrix for C(D',5P’) is 1 1 1 0 30 0 10 1+ 42° + 162* + 32z° + 402° 4 3227 met pe X— we Oo eH ow = t2 G2
The weight enumerator of C(D’,5P’) is then and the weight enumerator of C*(D’,5P’) is
1 +427 + 1223 + 722% + 17625 + 2322 + 12827,
In particular, the minimum distance of C*(D, 5P) is 3, while the minimum distance of
C*(D', 5P’) is only 2
REFERENCES
[1] E Arbarello, M Cornalba, P.A Griffiths, J Harris, Geometry of algebraic curves, Volume I, Springer-Verlag, New York, 1985 [2] A Garcia and P Viana, Weierstrass points on certain non-classical curves, Arch Math 46 (1986), 315-322 [3] V D Goppa, Algebraico-geometric codes, Math USSR-Izv 21, no 1, (1983), 75- 91
[4] V D Goppa, Geometry and codes, Kluwer, Dordrecht ,1988
[5] J H van Lint, Introduction to coding theory, Springer-Verlag, New York, 1982
{6} J H van Lint and G van der Geer, Introduction to coding theory and algebraic ' geometry, Birkhauser, Basel,1988
[7] G Oliveira, Weierstrass semigroups and the canonical ideal of non-trigonal curves, Manus Math 71 (1991), 431-450
[8] F-K Schmidt, Zur arithmetischen Theorie der algebraischen Funktionen II Allge-
meine Theorie der Weierstrasspunkte, Math Z 45 (1939), 75-96
{9] H Stichtenoth, A note on Hermitian codes over GF(q"), IEEE Trans Inform Theory 34, no 5 (1988), 1345-1348, {10] K.-O Stéhr and J.F Voloch, Weierstrass points and curves over finite fields, Proc London Math Soc (3), 52 (1986), 1-19 [11] H.J Tiersma, Remarks on codes from Hermitian curves, IEEE Trans Inform The- ory, IT-33 (1987), 605-609 [12] K Yang and P.V Kumar, On the true minimum distance of Hermitian codes, these proceedings
On a Characterization of Some Minihypers in PG(tq) (q=3 or 4) and its Applications to Error-Correcting Codes
Noboru Hamada! Tor Helleseth?
Department of Applied Mathematics Department of Informatics Osaka Women’s University University of Bergen
Sakai, Osaka, Japan 590 N-5020 Bergen, Norway’
Abstract
A set F of f points in a finite projective geometry PG(t, q) is an {f,m;1, q}—minihyper if m (> 0) is the largest integer such that all hyperplanes in PG(t,q) contain at least m points in F where t > 2, f > 1 and q is a prime power Hamada and Deza [9], [11]
characterized all {2vq41 + 20ø+1, 20a + 2ug;t,q}—minihypers for any integers ¢, 4a and
6 such that g > ð and 0 < a < Ø < ‡ where tœị = (q@'— 1)/(q— 1) for any integer I> 0 Recently, Hamada [5], [6] and Hamada, Helleseth and Ytrehus [18] characterized all {20 + 20a, 2uo + 201;†, g}—minihypers for the case ¢ > 2 and q € 13,4} The purpose of
this paper is to characterize all {2vq41 + 2ug41,2va + 2ug;t, 4}—minihypers for any integers
!,q;œ and Ø such that g € {3,4}, 0 < œ < Ø < ‡ and Ø # œ-+ 1 using several results in Hamada and Helleseth [12], [13], [14], [16], [17]
1 Introduction
Let F be a set of f points in a finite projective geometry PG(t,q) of ¢ dimensions where t > 2 and q is a prime power If |F M H| > m for any hyperplane H in PGi, 4) and |F 1 H| = m for some hyperplane H in PG(t,q), then F is called an { f, m;t, 4}—minihyper where m > 0 and |A| denotes the number of elements in the set A In the special case ¿ = 2
and m > 2, an {f,m;2,q}—minihyper F is also called an m—blocking set if F contains
no 1—flat in PG(2,q) The concept of a minihyper (or a min*hyper) has been introduced by Hamada and Tamari [19]
1 Pantally supported by Grant-in-aid for Scientific Research of the Ministry of Education, Science and Culture under Contract Numbers
403-4005-02640182
Trang 26In the case k > 3 and 1 < d < g*-! — g, d can be expressed uniquely as follows:
h d= gt! - ¢ + 3z) ` t=l
using some ordered set (€, 1, ga; - y #„) in U(k — 1,q) where U(t,q) denotes the set of all ordered sets (€, 41, tạ, - ,g„) Of integers e,h and jy; such that ()0 <Se <g—1, 1 << (†—1)(qg— 1), 1 < gì < tạ < - < tụ < £ and (b) at most q — 1 of the ;’s
take the same value
Hamada [4] showed that in order to characterize all [n, k, d; g]—codes meeting the Grismer
h
bound (cf [2], [23]) for the case k > 3 and d = gẺ—1 _ ( + Yq" | it is sufficient to
i=1
solve Problem 1.1 below The connection between codes meeting the Griesmer bound and minihypers is explained in detail in Appendix IV
Problem 1.1 (1) Find a necessary and sufficient condition on an ordered set h Ằ
(€,#1,2,-++, a) in U(t,q) such that there exists a {en + DY my41,600+ + muita}
t=1 s=1
minihyper where v; = (q' —1)/(q —1) for any integer J > 0
h b
(2) Characterize all { E0 + 3 9,+1,E00 † 3 0ụạ; ta} ~ minihypers in the case where i=1 t=1 there exist such minihypers "
_ Let A(t,q) be the set of all ordered sets (A1,A2,° ,Ayg) of integers 4 and
Ai (¢ = 1,2, +,9) such that 1 < 9 < f{g— 1), 0 S Ai < Às¿ < - < dy <t-1,
A, # 0 and 0 < ni(A) < g-1 for! = 0,1, -,ý — 1 where n;(A) denotes the num-
ber of integers A; in A = (A1,A2,+++,A,) such that A; = 1 for the given integer | Note that there is a one-to-one correspondence between the set U (t,g) and the set A(t,q) as fol-
lows : (i) in the case e = 0, 7 = A and X; = Hi for i = 1,2, -,h and (ii) in the case E#0,1 =e+h, Ài = À¿ = - = À,¿ = 0 and Acti = pi for i = 1,2, -,k As occa-
sion demands, we shall consider the following problem instead of Problem 1.1 Of course, Problem 1.2 is equivalent to Problem 1.1,
Problem 1.2 (1) Find a necessary and sufficient condition on an ordered set 1 2
(A1,A2,°+-,Ay) in A(é,g) such that there exists a { >>» tàiSg}— minihyper i=1 = 1 1 (2 Characterize all {x ĐÀ; » tjt:4 }—minhypen in the case where there exist such minihypers - s=1 ”
Problem 1.2 was solved completely by Helleseth z1 in the case g = 2 and by Hamada [3] in the case g 2 3, 1 < r < ‡ and Ö < Àt < À2 < ‹ < À„ <£—1, Hence it is sufficient to solve Problem 1.2 for the case q > 3, 9 2 2 and A; = À¿ for some distinct integers i and j
In the case n = 2 and A1 = Ao, Problem 1.2 was solved by Hamada [7] In the case
7 = 3, Problem 1.2 was solved completely by Hamada [6], [7], Hamada and Deza [8] and
Hamada and Helleseth [12]-[17] In the case 9 = 4, g > 5 and Ay = Ae < Ag = Ag, Problem 1.2 was solved by Hamada and Deza [9], [11]
Recently, Hamada, Helleseth and Ytrehus [18] solved Problem 1.2 for the case 7 = 4, q € {3,4} and (Ai, A2, A3,A4) = (0,0,1,1) (cf Propositions 77.3 and ITT.4 in Appendix)
The purpose of this paper is to solve Problem 1.2 for the case 7 = 4, g € {3,4},
Ai = Az < Az = Aq and Az # Ai +1 using the results in Appendices I, II and II The main result is as follows
Theorem 1.1, Let t,q,a and @ be any integers such thatqg = 30r4,0<a<B<t
and đ # œ + 1
(1) In the case t < 26, there is no {20z+t + 204,1, 20„ + 20g;‡,g}— minihyper
(2) In the case t > 26+1, Fisa {2va41 + 20841; 20a + 2ug;t, q}— minihyper if and
only if F is a union of two a—flats and two 6—flats in PG(t,q) which are mutually disjoint Remark 1.1 Hamada and Deza [9], [11] showed that in the case g > 5 Theorem 1.1
holds for any integers ý,œ and § such that 0 < a < @ < t But Hamada, Helleseth and
Ytrehus [18] showed that Theorem 1.1 does not hold in the case g € {3,4}, a = 0 and 8 = 1 From Remark J.3 in Appendix I and Theorem 1.2 in Hamada and Helleseth {1141
it follows that Theorem 1.1 does not hold for any integers t,q,a and @ such that ¢ = 4, a>0,8=a+1 andt > 2042
Corollary L1 Let n = vj ~ 2ve41 — 2ugyi and d = g'-! — 29% — 2¢8 where q = 3
o4,0<a<f<tandB#at+l
(1) In the case k < 2 + 1, there is no [n, k, d; q]—code meeting the Griesmer bound (2) In the case k > 28 + 2, C is a [n,k,d;q]—code meeting the Griesmer bound if and only if Ở is congruent to some [n, k, đ; g]—code constructed by using a union of two a—flats and two #—flats in PG(t,q) which are mutually disjoint
Remark 1.2, It is unknown whether or not there exists a (93, 5,61; 3]—code meeting
the Griesmer bound Since n = 93,d = 61,0, = 1,v3 = 13,0s = 121 in the case
= 3,k = 5,a = 0 and Ø = 2, Corollary 1.1 shows that there is no [93, 5, 61; 3]—code meeting the Griesmer bound
2 The proof of Theorem 1.1 ,
Let F(a, 8,7, 6; t, q) denote the family of all unions [J V; of an a—flat Vi, a B—flat Vo, a
=]
y—flat Vs, a 6—flat V4 in PG(t, q) which are mutually disjoint whereO<a<B<7< 8 <t In order to prove Theorem 1.1, we prepare the following two theorems whose proofs will be given in Sections 3 and 4 respectively
Theorem 2.1 Let vp = 0, vj = 1, vp = q¢1 and v3 = q’-+9¢+1 where g =3 or 4,
Trang 27(2) In the case ¢ > 5, F € F(0,0,2,2;t,¢) for any {2v, + 2v3, 2v9+ 29; t, g}—
minihyper F
Theorem 2.2 Let ¢,@ and q be integers such that t > 20 > 6 and q=3or4 If Fisa {2v2 + 2ug41,201.+ 2vg;t,q}— minihyper such that (a) |FNG| = 2vg_; for some
(t — 2)—flat G in PG(t,q) and (b) FN H; € F(0,0,8 —1,@ ~1;t,q) for any hyperplane H; (1 < 7< q+1) in PG(t,q) which contain G, then F € #(1,1,8, 8; ‡,q)
Remark 2.1 Since Z{(1, 1, đ, đ; ý, g) = 0 in the case t = 28 (cf Remark I 1), Theorem 2.2 shows that in the case t = 2, there is no {2v2 + 2vg41, 201 + 20a; t,q}— minihyper F
which satisfies the conditions (a) and (b) in Theorem 2.2
(Proof of Theorem 1.1) It follows from Proposition J.1 and Remark J.1 that if F € F(a,a,8,f;t,q) in the case t > 28 +1, then F is a {20041 + 20g+:, 20a + 20g; ‡, g}— minihyper
Conversely, suppose there exists a {2va41 + 2vg41,2vq + 2vg;t,q¢}— minihyper F for
some integers t,q,a and @ such that g = 3 or 4,0 < œ< 8< tand 8 # œ+1 Then it
follows from Proposition I.2 ((i) in the case a = 0,¢ = 2, h = 2 and H1 = Hạ = Ö and
(Ù) in the case œ # Ú, e = 0, h = 4, pi =o =a and 43 = ạ = /đ) that there exists a (t — 2)—flat G in PG(t,q) such that |FNG| = 2vq—-1 + 2ug_1 where v_1 = vp = 0 Let
H; (j =1,2,-+-,q+1) be q¢ +1 hyperplanes in PG(t,q) which contain G
Case I : (a=0 and 8 =2) It follows from Theorem 2.1 that Theorem 1.1 holds
Case Il : (a=0 and B23) It follows from Proposition J.2 that FN H; is a
{5; + 2ug,2vg_1;t,q}— minihyper in H; for j = 1,2, -,q +1 where the 6; are non-
l + q+]
negative integers such that » 6; = 2 Without loss of generality, we can assume that either 1
(a) 61 = dg = - = b9~1 =0 and 8g = 6941 = 1 or (bì ối = 6a =-.- = q = O and 6541 = 2
(A) In the case 6) = 6 = - = 1; = 0 and 6 = q+1 = 1, it follows that FN H; is a {2ug,2vg_1;t,q¢}— minihyper in the (t — 1)—flat H; for i = 1,2, -,g — 1 and 'n H;
is a {v1 + 2ug, v9 + 2vg_1;t,q}— minihyper in H; for j = q,qg+1 From Remark J.2,
Proposition 5 , II.1 and TTT.1 in Appendices, it follows that (i) in the case t—1 < 2(Ø—1),
there is no {2ug, 2ug_1;t,q}— minihyper in any (t — 1)—flat H, a contradiction, and (ii) in the case ý — 1 > 2(đ — 1) +1, nH; c #(8~1,8— 1;t,q) for ¡ = 1,2, -,qgT— 1
and FN H; € F(0,8-—1,8—1;t,¢) for j = q,q+ 1 Hence it follows from (i) and Proposition I.3 (¢ = 2, h = 2, 41 = po = B) that (1) in the case t < 26 — 1, there is no
{20 + 20g41,2u9 + 2ug;t, q}— minihyper F which satisfies the condition (a) and (2) in the
case t > 28, F € F(0,0, 8, 8;t,q) Since F(0,0, 8, A;t,q) 4 0 if and only ift > 26+-1 (cf Remark 1.1), there is no {20 + 2vg+41,2v9 + 2vg_1;t,q}— minihyper in the case t = 28
(B) In the case &; = ốy = = q = O and 6,41 = 2, it follows from Remark J.2, Proposition I.5, Case I and induction on # that (i) in the case t — 1 < 2(8 — 1), there is no
{2vg, 2vg_i;t, q}— minihyper in #1, a contradiction, and (ii) in the case t—1 > 2(8—1)+1, +#'n51; e 7(8 —1,8 — 1;t,q) fori = 1,2, - ,ø and FN Hg+1 € F(0,0,8 — 1,8 — 1;‡,q) Hence it follows from (i), Proposition I.3 and Remark I.1 that (1) in the case t < 2, there is no {201 + 2ug41,2v9 + 2vg;t,q}— minihyper F' which satisfies the condition (b) and (2) in the case t > 28+1, F c 7(0,0,6,6;t,g)
From (A) and (B), it follows that Theorem 1.1 holds in Case II
Case II : (a=1 and Bz3) It follows from Proposition I.2 (¢ =0,h = 4, mì = tạ = 1,
U3 = tạ = B) that FN A; is a {2v1 + 2vg,2v0 + 2vg_i;¢,q}— minihyper in Hj for
j = 1,2, -,qg +1 Hence it follows from Remark J.2, Cases I and II that (i) in the
case ý — 1 < 2(Ø — 1), there is no {2v; + 2ug,2v9 + 2vg_1;t,q}— minihyper in Hj, a
contradiction, and (ii) in the case t~—1 > 2(8 — 1) +1, FAH; € F(0,0,8 —1,8 — 1;t,¢)
for j = 1,2, -,q+1 Using Theorem 2.2 and Remark J.1, it can be shown that Theorem
1.1 holds in Case III -
Case IV ; (az2 and Bzat2) It follows from Proposition I.2 that FN Hj is a
{2ta + 2ug, 2vq—1 + 2ug_1; t,q}— minihyper in H; for j = 1,2, -,qg+1 Hence it follows from Remark J.2, Cases I, II, HI, induction on @ and §, Proposition I.3 and Remark J.1 that Theorem 1.1 holds in Case IV This completes the proof of Theorem 1.1
3 The proof of Theorem 2.1
Let #(Ài,Àa, - ; Àn;‡,g), Z(0,1,1;2,3), Z2(0,0,1,1;£,3) and Z;(0,0,1,1;£,4) (¿ =
1,2,3,4) denote the families given in Appendix I, Defnition II.1, 11.2, TTT.2 respectively In order to prove Theorem 2.1, we prepare the following four lemmas whose proofs will be given in Sections 5, 6, 7 and 8 respectively
Lemma 3.1 In the case ¢ > 4, there is no {2u -+ 2u, 2u -+ 20a;t,3}— minihyper F
such that (a) |F N G| = 2 for some (t— 2)—flat G in PG(t,3) and (b) FN Hi € F(1,1;2,3),
n1 c F(1,1;,3), FN Hs € F(0,1,1;t,3) and FN Hy € F(0,1,1;t,3) where vp = 0,
v1 = 1, ve = 4, v3 = 13 and the H; denote four hyperplanes in PG(t,3) which contain G
Lemma 3,2 In the case ý > 4, there is no {2v1 + 23, 2v9 + 2v2;t,3}— minihyper F
such that (a) |F' G| = 2 for some (t — 2)—flat G in PG(t, 3) and (b) FN Ai € F(1,15;#,3),
F He € F(1,1;t,3), FN Hg € F(0,1,1;t,3) and FN Hy € F(0,1,1;t,3) where the H;
denote four hyperplanes in PG(t,3) which contain G
Lemma 3.3 In the case ¢ > 4, there is no {2v; + 23, 2v9 + 2v2;t,3}— minihyper F such that (a) |F'M G| = 2 for some (t — 2)—flat G in PG(t,3) and (6) FN H; € F(1,1;#,3),
FO Hp € F(1,1;1,3), FN Hs € F(1,1;t,3) and FA Hs € Fo(0,0,1,1;t,3) for some @ in
{1,2,3,4} where the H; denote four hyperplanes in PG(t,3) which contain G
Lemma 3.4 In the case t > 4, there is no {20 + 20x, 20a + 20a;t,4}— minihyper F such that (a) [7 N G|.= 2 for some (¢ — 2)—flat G in PG(t, 4) and (b) f'n.H; € 7(1,1;,4)
for i = 1,2,3,4 and FM Hs € Fo(0,0,1,1;#,4) for some @ in {1,2,3,4} where vp = 0,
v1 = 1, v2 = 5, v3 = 21 and the H; denote five hyperplanes in PG(t,4) which contain G
(Proof of Theorem 2.1) Suppose there exists a {2 + 2uạ, 2uo + 2ua; ,g}— minihyper
Trang 28that (a) [FNG| = 2 for some (t ~ 2)—flat G in PG(t,q) and (b+) FN dH; is a {6 + 292, 201; ¢, 9}~—minihyper in 4H; for any hyperplane H; (1 < i <q+1) in PG(t,q) + which contain G where the 6; are nonnegative integers such that > 6; = 2 Without loss of generality, we can assume that either (a) 6) = dg = + = fyi = 0 and & = 641 =1
or (8) 61 = 2 = ++» = & = 0 and 641 = 2 TS
Case I: (q=3, 6: = 62 = 0 and 63 = &4 = 1) It follows from Remark I -2, Propositions 1.5 and II.2 that (i) in the case ý — 1 = 9, there is no {2u¿, 201; ¢,3}—minihyper in Hi, a contradiction, and (ii) in the case — 1 > 3, n Hị F(1,1;2,3), FN He € F(1,1;t,3)
and either FN H; € #(0,1,1;‡,3) or FN A; € F(0,1,1;#,3) for i = 3,4 It follows from Lemmas 3.1 and 3.2 that FN H; € F(0,1,1;#,3) for i = 3,4 Hence it follows from
(), Proposition 1.3 (e = 2, h= 2, mị = Ho = 2) and Remark J.1 that Theorem 2.1 holds
in Case I
Case IT : (q=4, 6; = 69 = 63 = 0 and 64 = ốg = 1) It follows from Remark
I.2, Propositions I.5 and ITI.1 (a = 0,8 = 7 =1) that () in the case ‡ — 1 = 2, there
is no {2ve, 201; t, 4}—minihyper in 4, a contradiction, and (ii) in the case t — 1 > 3, FO F; € F(1,1;t,4), for i = 1,2,3 and FN Hj € F(0,1,1;t,4) for j = 4,5 Hence it follows from (i), Proposition I.3 and Remark J.1 that Theorem 2.1 holds in Case IL
Case III : (q=3, 6, = 59 = &3 = 0 and &4 = 2) It follows from Remark J.2, Propositions 1.5 and IJ.3 that (i) in the case t— 1 = 2, there is no {2v2, 20; ¢,3}—minihyper in Hj,
a contradiction, and (ii) in the case t- 1 > 3, FO A; € F(1,1;t,3), fori = 1,2,3 and either FN Hg € F(0,0,1,1;t,3) or FN Hy € Fo(0,0,1,1;t,3) for some @ in {1, 2,3, 4} Hence it follows from (i), Lemma 3.3, Proposition J.3 and Remark J.1 that Theorem 2.1
holds in Case III
Case IV : (q=4, 6: = 82 = 53 = 64 = 0 and 65 = 2) It follows from Remark 1.2,
Propositions 1.5 and TIT.4 that (¡) in the case t—1 = 2, there is no {2v2, 2; ¿, 4}—minihyper
in 4, a contradiction, and (ii) in the case 1-1 > 3, FN A; € F(1,1;t,4) for i = 1,2,3,4
and either FN Hs € F(0,0,1,1;,4) or FAHs € Fo(0, 0, 1,1; #,4) for some 6 in {1,2, 3,4}
Hence it follows from (i), Lemma 3.4, Proposition J.3 and Remark J.1 that Theorem 2.1 holds in Case IV This completes the proof of Theorem 2.1
4 The proof of Theorem 2.2
Lemma 4.1 Let t,6 and q be integers such that ¿ > 2/ + 1 > Tandq> 3 If Fis a {2+2 + 20gø+1,20 + 2ug;t, q}—minihyper such that F = X UY; U Yo for some set _X (of
2v2 points in PG(t,q)) and some /Ø-flats Yi, Yo in PG(t,q) which are mutually disjoint, then X € F(1,1;t,¢) and F € F(1,1, 8, B;t,q)
Proof If |X N H| > 2 for any hyperplane H in PG(t,q) and |Xn H| = 2 for some hyperplane hyperplane H in PG(t,q), it follows from |X| = 2v2 and Proposition 5 that 2
X € F(1,1;i49) and F € F(1,1,8,54,4) Sinee |F n BỊ = |X n BỊ + 3} JY; n BỊ and a IY H| = vg or vgy1 for any hyperplane H in PG(t,q), it is sufficient to show that there is no hyperplane H in PG(t,q) such that [FN H| = vg + vgy1, 1 + vg + vg4i, 2g41 or 1 + 20g+1 / - Suppose there exists a hyperplane H in PG(t,q) such that |FN A| = 3 + 2ua, 4 + 20g; 9ø + 9g8+1y Ì + 08 + vg41, 2g41 Or 1 + Qugii
Case I: (\F 1 H| = 3v; + 2vg) Suppose there exists a (¢ — 2)—flat G in H such that |FNG| < 2»g_t Let H; (¡ = 1,2, - - -, ạ) be q hyperplanes in PG(¢, q), except for H, which contain G, Since |F| = 2v2 + 2vg41 and |F N H;| > 2v1 + 2vg for i = 1,2, -,g, it follows
g
- from quvg.1 = vg — 1 and qua = 0ø+ — 1 that Fl = |FO Al + >) {JF Bil —|F NGI} >
t=1
2ug41 + 2q +3 > |F|, a contradiction, Hence |F N G| > v; + 2vg_1 for any (¢ — 2)—flat G in H, Using Proposition 6 (8 = t —1,€1 = 1,eg-1 = 2), we have [FN A| > v2 + 2vg > 3+ 2vg = |F NA, a contradiction Therefore, there is no hyperplane H in PG(t,q) such that [FN H| = 3 + 2ug
Case II : (qz4 and |F 1 H| = 4v; + 2vg) Using a method similar to Case I, it can be shown that there is no hyperplane H in PG(t,q) such that |FM H| = 4+ 2ug in the
‘case g > 4
Case III : (q=3 and|F 1 H| = v2+2vg) Suppose there exists a (t—2)—flat G in H such that |F 1 G| < -1+v1+2vg_1 Let Hj (i = 1,2, 3) be three hyperplanes in PG(t, 3), except
3 -
for H, which contain G Then |F| = |#'n H|+ 3> {|f'n Hil -|F NG} = 22a+i+a+6 >
«=
|F|, a contradiction, Hence |FNG| > v1 + 2vg_; for any (t — 2)—flat G in H Since [FN H| = v2 + 2vg, it follows from Proposition I.6 that there exists a (t — 2)—flat G in H such that |FG| = v1 + 2vg_1 Let Oj (i =1,2,3) be three hyperplanes in PG(t,3),
except for H, which contain G
3
Sinee 3 |EƑn(H\G)| = |F - |fnHỊ = 3(2-32—! +1) + 1 and |Ƒn(H\G)| = iz]
|F NU; |-[F NG] > 2-3°-1 41 for é = 1,2,3, there exists a hyperplane Il in {11,, 12,3}
such that [F' n HỊ = (20 + 2uø)+1 = 3+20, a contradiction Hence there is no hyperplane
# in PG(t,3) such that |f'n HỊ = 4 + 20a ỉn the case g = 3
Case IV: (|F 1 H| = vg + vg41) Using a method similar to Case Ill, it can be shown that there exists a (t—2)—flat G in H such that |F N G| = vg_i+vg Let i; (i = 1,2, -,g) be q hyperplanes in PG(t,q), except for H, which contain G
Since 3° JF (H:\G)| = |F| - [FOB = s(g6~!+3) + 2 and |Fn(H\G)| = i=l
lFNO,| - j#n Gl > 98-142 fori = 1,2, -,9, there exists a hyperplane I in
{IHI,Hạ, - - -, Hạ} such that |FNO| = 3 + 2og or 4 + 2g, a contradiction Hence there is no hyperplane H in PG(t,q) such that |FN H| = vg + vga
Similarly, it can be shown that there is no hyperplane H in PG(t,q) such that |FN #| =
20g41
Case V: (FN H| = 1+ vg + 9g+¡ or 1 + 2vg41) Using a method similar to Case II, it can be shown that there exists a hyperplane II in PG(t,g) such that |FN H| = 3 +21a,
Trang 29(Proof of Theorem 2.2) Let F be any {2v2 + 2g1,2v1 + 2vg;t,q}— minihyper which
satisfies the conditions (a) and (b) in Theorem 2.2 where t > 28 > 6 Then FN H; = {Pi, Pi2} U Wa U We for some points Pi, Pj and some (@ ~ 1)—fiats Wa, Wie in H;
which are mutually disjoint for i = 1,2, -,¢+1 Since |GMW| = vg_1 or vg for any
(6 — 1)—flat W and any (t — 2)—flat G in Hj, it follows from |FNG| = 20g—1 that h„ ø G and GnWwiy = V; (¡ =1,2, ',g+1, i= 1,2) for some (8 — 2)—flat V; in G
a+ 4 4
Let X = U {Pa,Pi2}, Vì = U Wy and Yo = U Wy
t=1 i= =
Let E; = Hy (Wii © Wo1) for t = 3,4, -,¢+1 Then E; is a (@ — 1)—flat in H; such that GN E; = Vi fori = 3,4, -,qg+1 Note that (i) Wir U Wai U Bs U -U Bayt isa
f—flat in PG(t,q) and (i) either Wi = Ej or Wii Eị = Vi for each 7 Hence if Wy = EF;
for i = 3,4, -,q+1, then Yi is a 6—flat in PG(t,q)
Suppose Wj, E; = V, for some 7 in {3,4, -,q+ 1} Without loss of generality,
we can assume that Wo; = Eo, Wa, = Bs, -, Wo-1,1 = Eạ.¡ and We NEg =e = Wo41,1 9 Eq4i = Vi for some @ in {3,4, -,¢+1} where Bo = War
Let 2 be a (¢ — 2)—flat in Hy41 such that Fyy1 C2, ON Wo4i1 = Vi and ANW isa
(8 — 3)—flat in G Let 1, (J = 1,2, -,g) be q hyperplanes in PG(t,q), except for H,+1,
which contain 2 Then 1) 9 W411 = 20 Wo41,1 = Vi and I, N Wiz is a (8 — 2)—filat in
A; (ie [ri Wia|l = vg_1) for 1 = 1,2, -,g andi = 1,2, -,q+1
Since Wi, C Da, Wai C Hạ and Wịi C Hạ, for some integers a, 93, 04,°++ 54 in
{1,2, -,q}, there exists a hyperplane II in {H, Hạ, - - -, Hạ} such that TN Wa = Vi for t= 1,2, -,g Hence it follows from ` = X UY+U2 that |#'n HỊ = |Xn HỊ+|#⁄+n HỊ+
Yan] = |X) + [Wi| + (a+ 1)»a—¡ = |X nHỊ + (q+ 2)vg_1 < 20; + 2ug unless
g = 3,8 = 3 and |XnHỊ = 8
Case Ï : (qz4 or q=3 and 8z4) \t follows that |EF'n THỊ < 2v; + 2ug, a contradiction Hence Wi = E; for i = 3,4, -,q +1 This implies that Y; is a @—flat in PG(t,q) Similarly, it can be shown that Y2 is a G—flat in PG(t,q) Hence it follows from Lemma
4.1 that F € F(1,1,6,8;t,q) in the case ¢ > 2Ø + 1
In the case t = 28, there do not exist two 6—flats Y; and Y2 in PG(t,q) such that
YiNYo = 0 Hence in the case t = 28, there is no {2v2 + 2ug41,2v1 + 2ug;t, q}—minihyper
F which satisfies the conditions (a) and (b) in Theorem 2.2 This implies that Theorem 2.2 holds in Case 1
Case If : (q=3 and 8=3) It follows that @ = 3 or 4, ie, either (a) W31N Bs = Wai Ey = Vị or (@) Wa, = Es and Wa N Ey = Vi
(A) In the case W3; = E3, we can assume without loss of generality that Wi1 C Th, Wai C Wi and Ws: C 11 Hence there exists a hyperplane II in {Hạ,IIạ} such that Pir ¢ and 1N Wa = Vi fort = 1,2,3 This implies that [XII] < 7, ie, |F OO} < 2v; + 2vg, a contradiction Using a method similar to Case I, it can be shown
that Theorem 2.2 holds in the case (A)
(B) In the case W31N £3 = Vj and P3, € Es there exists a hyperplane Il in {I;, U2, Hạ} such that Ps; ¢ II and IN Wy = V; fori = 1,2,3 This implies that |X nHỊ < 7, ie,
|\F NT < 2v; + 2ug, a contradiction Hence Theorem 2.2 holds in the case (B)
tot
(C) In the case W31M E3 = V; and Ps; ¢ E3, let 2 be a (¢ — 2)—flat in H3 such that Psi ¢ 2, By C2, QN Wai = V; and NN Ve is a (8 — 3)—fiat in G Let (1 = 1,2,3) be three hyperplanes in PG(?, 3), except for H3, which contain Q Since Ps; ¢ 0; for! = 1, 2,3, there exists a hyperplane II in {1j, Ip, 113} such that [FN 0] < 2v; + 2vg, a contradiction Hence Theorem 2.2 holds in the case (C) This completes the proof of Theorem 2.2
5 The proof of Lemma 3.1
Suppose there exists a {2v; + 2v3, 2vp + 2v9;t,3}—minihyper F which satisfies the conditions (a) and (b) in Lemma 3.1 Then FNG = {Pi, Po}, FN A, = Ly U Ly,
FO He = La U Loe, FN Hz = £3, U L32 U {P3} and FN Hy = V\{Q1, Q2, Q3, Qa} for some 2—flat V in H4 and some 4—arc {Qi, Q2,Q3, Qa} in V where Lj and Lj» are 1—flats in H; such that Ly N Le =0,GN Ly = {Pi} and GN Lye = {P2} fori = 1,2,3 and Ps is
a point in H3\G and GN V is a 1—flat in G which contains two points P\ and P› Without loss of generality, we can assume that GN.V = {P1, Po, Q1, Qo}
In order to prove Lemma 3.1, it is sufficient to show that there exists a hyperplane I in PG(t,3) such that |F N1| < 2ve = 8 Let B; = Hin (in @ D3) fori = 1,4 Then
£; is a 1—flat in H; such that GN E; = {Pi} for i = 1,4 Note that either Li; = EF, or
tị n Eị = {Pi}
Case I: (uN Ey = {Pi}) Let M = GN(Li@F;) Then M is a 1—flat in G passing through P, Let 2 be a (¢ — 3)—flat in G such that 2 M = {Pi} and Pp ø Ð Let H;(7 = 1,2,3) be three hyperplanes in PG(t, 3), except for Hi, which contain the (t—2)—fiat
2@® £; in Mi Since Po € V, Po € Liz, Pa # H; and M ¢ I; fori =1,2,3 andj = 1,2,3,
we have | n TĨ;| = 4, |L¿ n H;| = 1 and Lun Ij = {Pi} fori = 1,2,3 and 7 = 1,2,3
Since Lại C H„, Dại C H„ and Py € Hs for some integers a and Ø in {1,2,3},
there exists a hyperplane II in {1,112,113} such that P; ¢ 0 and IN Li = {Pi} for i = 2,3 This implies that there exists a hyperplane II in PG(t,3) such that |FOO| = 3
, |t¿n HỊ + J(°n ạ)n HỊ < 3+ |Vn HỊ = ï < 2ua, a contradiction Hence Ly, = Fx Case IT: (Li, = Ey) Let 3 be a (t — 3)—flat in G such that P; € © and Py ZX
Let I; (j =1,2,3) be three hyperplanes in PG(t,3), except for H4, which contain the (t — 2)—flat © @ Ea in Hy Since Ly) C I, Ln C Ta, £31 C Wa and P3 € Ig for some integers œ and / in {1,2,3}, there exists a hyperplane II in {H¡, Hạ, Hạ} such that Ps ¢ I 3 and IA Dix = {Py} for i = 1,2,3, ie, FAD] = ¥ [bạ n THỊ + |('n Hạ) nHỊ < 2m, i=l
a contradiction Hence there is no {20 + 2u, 2uạ + Dun; t,3}—minihyper F which satisfies
the conditions (a) and (b) in Lemma 3.1
t=
6 The proof of Lemma 3.2
Trang 30
some 4—are {Qj1, Qj2,Q53, Qja} in V; (j = 3,4) where Ly and Lj are 1—flats in H; such
that Da N Lye = 0, GN Ly = {Pi} and GN Ly = {F2} for i = 1,2
Since GV; is a 1—flat in G which contains two points P, and Pe, we can assume
without loss of generality that GN V3 = GN V4 = {Pi, Pe,Qi,Q2} where Qj = Qi and
Qj2 = Qe for j = 3,4 Since {Q1,Q2,Q43, Qa4} and Pị @ P¿ are a 4—arc and a 1—flat, tespectively, in the 2—flat V4, it follows that (Pi @ Pa) n (Q43 ® Qa4) = {Pi} or {Po} Without loss of generality, we can assume that (Pi @ Pe) (Q43 © Qas) = {Pi}
Let 2 be a (¢ — 3)—flat in G such that P, € 5 and Po ¢ X Let II; (1 = 1,2, 3) be three
hyperplanes in PG(2,3), except for H4, which contain the (¿ — 2)— fat 5 @ (Q44 ® Q44) in
Hạ Then |Ji+fñ Hụ| = 1, |Zz¿ 1| = 1, [Vạn Tĩị| = 4 and W+n 1; = {Q43, Q44, Pi, R}
(ie (FM V4) N1y| = 2) for 1 = 1,2,3 where R denotes the point in Q4¿ @ Q44 except
for three points , Q4; and Q44
Since Li; C H„ and Lại C Wg for some integers a and đ in {1,2,3}, there exists
a hyperplane H in {H\,Hạ,Hạ} such that Hn Eịi = {P(} and H0 Lại = {PP} ie, |ƑnHI = |Ei¿n HỊ + |Ez¿n HỊ + J(Œ“n W)n HỊ + (n4) HỊ —1 < 7 < 2x, a
contradicion Hence there is no {20 + 2uạ, 2o + 2a; ý, 3}—minihyper F' which satisfes the conditions (a) and (b) in Lemma 3.2
7 The proof of Lemma 3.3
Suppose there exists a {20 + 20, 2o + 2u2;‡,3}—minihyper F which satisfies the conditions (a) and (b) in Lemma 3.3 Then FONG = {P1, Po}, FN A, = Li U Ly,
FO He = La, U Loo, FN Hs = £31 U £32 where Ly and Lie are 1—flats in Hj such that tị n Lạ = 0, GN Ly = {Pi} and GN Ly = {Po} for i = 1,2,3
CaseI: (8=1) It follows from Definition IJ.2 that FM H4 = V\{Q1, Qo, Q3} for some
2—flat V in H4 and some 3—are {Q1,Q2,Q3} in V Let Eq = H4N(L11 @ Lan) and let 3 be a (¢—3)—flat in G such that P; € â and Py  E Let 0;(j = 1,2, 3) be three hyperplanes in PG(t,3), except for H4, which contain the (¢ — 2)—flat 2 @ E4 in H4 Then there exists a hyperplane II in {H, Hạ, Hạ} such that |Ƒ'n HỊ < 29a, a contradiction
Case IT: (8=2) It follows that FN H4 = (V\S)U {P} for some 2—flat V in H4, some
4—-arc S = {Q1, Q2,Q3, Q4} in V and some point P in Ha\V Let Bạ = Hạn (bai @ bai) and Gf\W = {Pìi,P›, Qì, Q2}
(A) In the case P ¢ Ey let M = GN(E4 @ P) and let Ð be a (¿ — 3)—flat in G such that UNM = {Pi} and P ¢ B Let Ij (j = 1,2,3) be three hyperplanes in PG(t,3), except
for Hạ, which contain © © E4 Since P ø H; and |V n H;| = 4 for j = 1,2,3, there exists
a hyperplane II in {IH, Hạ, Hạ} such that |#' n HỊ < 2a, a contradiction
(B) In the case P € la, leét V = Gn (Ea @ Q3) and let D be a (t — 3)—flat in G such
that N C & and Pp ¢ X Let Il; (j = 1,2,3) be three hyperplanes in PG(t,3), except for
H4, which contain 5 @ Eạ Since VN; = P, ® Qs (i.e., |(V\S) NO] < 3) for j = 1,2,3,
there exists a hyperplane II in {H¡, Hạ, Hạ} such that |#' n HỊ < 2ve, a contradiction
Case III : (8=3) It follows that F Hy = LU K* for some 1—flat L in H4 and some
minihyper K* in F(0,0,1;¢,3) such that LNK* = 9, GNL = {Pi} and GNK* = {P)} Let
V be the 2—flat in H4 which contains K* and let M = GNV Let Eq = Hạn (bia © Loe)
and let & be a (¢ — 3)—flat in G such that P, ¢ Z and ON M = {P} Let 0; (j = 1,2,3) be three hyperplanes in PG(t,3), except for H4, which contain 2 @ Es
Since M Â â and |K*N L*| < 3 for any 1—flat L* in V, VNU; is a 1—flat in V and [K*N(VNO;)| < 3 for j = 1,2,3 Hence it follows #om |L¿+H;| = 1 and
|bnTI;[ = 1 (i =1,2,3, j = 1,2,3) that there exists a hyperplane II in {IH, Hạ, Hạ} such
that Ƒ'n HỊ < 2w, a contradiction,
Case IV : (6=4) It follows from Definition I7.2 that FM Hy = K for
some minihyper K in F4(0,0,1,1;t,3) where K = {(£0),(1),(2),(é3),(2é0 + £1),
(2&0 + &2), (2€0 + &3), (261 + €2), (2t + &3), (22 + &3)} for some four linearly independent
points (£0), (€1), (2) and (¢3) in PG(t,3)
Let W be the 3—flat in H4 generated by four points (to), (£1), (€2) and (¢3) in PG(t, 3) Since G is a (t -2)—flat in the (t—1)—flat H4 such that KNG =(FNH4)NG=FNG= {Pi, Po}, WONG must be a 2—flat (denoted by A) in G such that KNA = {P,, P2} Without loss of generality, we can assume that P; = (2f0 + 1) and P, = (2¢2 + 3)
Lett J = Pi OP, Ai = J @ (261 +6), Ao = J @ (fo), Az = JO (2) and Ag = J@ (fo + é1 + 2 + é3) Then A; (i = 1,2,3,4) are four 2—flats in W which contain
J and KM Ai = {(2€o + £1), (2€0 + €2), (2€o + 3), (261 + €2), (261 + @3),(2@ + é3)}, KN Aa = {(€o); (1), (260 + £1); (262 + &3)}, KM As = {(E2), (€3), (2Eo + £1), (2£2 + €3)}
and KM Ag = {(2é + &1), (2£2 + &3)} where A = Ag Let Eq = Hn (£11 @ Lại) Then
E4 is a 1—flat in H4 such that GM E4 = {Pi} Note that either By C W or EsnW = {Pi} (A) In the case Ey C W (ie., Eq C Aj) for some i in {1,2,3}), there exists a 2~flat V in W such that Ey C W, Po ¢ Vand |KNV| < 4 Let N = GNV Then N is a 1—flat in G such that P; € Ñ and P ý N
Let 5 be a ( — 3)—flat in G such that W C Ð and P; ø Ð Let H; (7 = 1,2,3) be three
hyperplanes in PG(t,3), except for Hạ, which contain Ð @ Es Since |L¿a\ H;| = 1 and
WñH; = V tor ¡ = 1,2,3 and j = 1,2,3, there exists a hyperplane II in {I1, Ilo, 113} such
that |#'n THỊ = 3+ |Kn VỊ < 7 < 2vo, a contradiction
(B) In the case Ey W = {Pi}, let V be a 2—fat in W such that Pị € V, Pa g V and |K NV| < 4 Let 2 be a (¢ — 2)—flat in H4 such that By CQ and NNW = V Let Il; (j = 1,2,3) be three hyperplanes in PG(t,3), except for H4, which contain 2 Since
Wol; =Wn02 =V for j = 1,2,3, there exists a hyperplane II in {H, Hạ, Hạ} such that
LF Il] < 2ve, a contradiction This completes the proof
8 The proof of Lemma 3.4
Suppose there exists a {2v, + 2v3,2vp + 2vo;t,4}—minihyper F which satisfies the
conditions (a) and (b) in Lemma 3.4 where vp = 0, 1, = 1, ve = 5 and vg = 21 Then FOG = {P,, Po} and FN H; = Ma U Mi for some 1—flats Mj; and Mjz in H; such that
Mi Mi = 0, GN Ma = {Pi} and GN Mia = {Py} for i = 1,2,3,4
Case I : (@=1), It follows from Definition IIJ.2 that FM Hy = K for some minihyper
Trang 31Then E is a 1—flat in Hs such that GN FE = {Pi} Note that (i) KNG =(FNHs)NG= FOG = {P,, P2} and (ii) either EC V or ENV = {Pi}
(A) In the case E C V, let 2 be a (¢ — 2)—flat in Hs such that QNV = E Let Tl, (1 = 1,2,3,4) be four hyperplanes in PG(t,4), except for Hs, which contain 2 Since
Po € Mig, Po € V, Po ¢ 2 and Py ¢ I; for i = 1,2,3,4 and J = 1,2,3,4, it follows that that |Mjo NT;| = 1 and VN; = VN = E Since Mii C Wa, Mai C Ila, Mai C UW, and
Ma C Il, for some integers a,b and c in {1,2,3,4}, there exists at least one hyperplane II in {th, Ho, Ms, Hạ} such that 1M Mj, = {Pi} fori = 1,2,3,4 This implies that IFNO| = > Man HỊ +|(tn ;) n HỊ < 4+|V n 0| = 9 < 2a, a contradiction Hence YnE= {Py} Note that 2v2 = 10 in Lemma 3.4
(B) In the case VN E = {Pi}, let 0 be a (¢ — 2)—flat in Hs such that EZ C Ô and QNV = N where N is a 1—flat in V such that P) € N and Po ¢ N Using a method similar to (A), we have a contradiction Hence 6 # 1
Case II : (0=2 or 3) Using a method similar to Case I, it can be shown that there exists a hyperplane II in PG(t,4) such that |F NII| < 2ve, a contradiction Hence @ # 2,3
Case Ill: (@=4) It follows from Definition II7.2 that FN Hs =
some 1—flat L in Hs and some minihyper K* in F(0,0,1;t,4) such that LQ K* =
where K* = {(wo),(w1), (wa), (wo + 1), (wo + we), (wi + we)(wo + wi + we)} for some
noncollinear points (wo), (wi) and (we) in Hs Note that |K*N E| < 3 for any 1—flat E in V where V denotes the 2—flat generated by the three points (wo), (w1) and (we)
Since |LNG| = 1 or 5 and (LUK*)NG = (FN Hs)NG = FONG = {Pi, Py}, we can assume without loss of generality that K* NG = {P,} and LNG = {Po} Let E = H;sn (Min @ Mai) Then either EC V or ENV = {Pi}
(A) In the case FE Cc V, let 0 be a (¢ — 2)—flat in Hs such that Py ¢ © and
QNV = E, Let UW, (l= 1,2,3,4) be four hyperplanes in PG(t,4), except for Hs, which
contain Q Then |Mi NT} = 1, [ENT = 1 and VN, = EF fori = 1,2,3,4 and J = 1, 2, 3,4 Hence there exists a hyperplane II in {H;,H;,Hạ,HẠ} such that
|FND| = 3 Men + |LrnHmỊ + |K*nE| <
YnE= {Pi}
(B) In the case VN E = {Pi}, let 2 be a (¢ — 2)—flat in Hs such that EB Cc © and QNV = N where N is a 1—flat in V such that P; € N and Po ¢ N Using a method similar
to (A), we have a contradiction Hence 6 # 4 This completes the proof of Lemma 3.4 8 < 2ve, a contradiction Hence
Appendix I Preliminary results in the general case
Let U(t,q) denote the set of all ordered sets (¢, 1, 2, - ; úy) Of integers e, h and m¿
( = 1,2, -,Ä) such that 0 < £ < g—1,1 < h < ((—1)(4—1), 1 < gì Spo S + S Hạ <†
and 0 < m(g) € qT— 1 for Ì = 1,2,-‹+,ý — 1 where n¡(ø) denotes the number of integers
Hị Ìn w = (oases +,#) such that y; = I for the given integer J In the case k > 3 and LU K* for 1<d< qg*-! — gq, d and the Griesmer bound can be expressed as follows d= q'-! -(+#z) and n> HE — =~ (+3 aun] t=1 i=1 (11) using some ordered set (€; /1; ga; - - - any integer | > 0
Let Fu(é,#1,42,-++,#ajt,¢) denote the family of all unions of e points,
a pi—Hat, a pe—flat, -, a pa—flat in PG(t,q) which are mutually disjoint where (€,/11,/12, -,#,) € U(t,q) As occasion demands, we shall denote Fule, 81; H3; ° * * › lạ; É, g) by /(Àu, Às,- “Àn; È, g) where ? = h+e, À¡ =0(¡= 1;2,: +56) and Aetj = yj (7 =1,2, -,h) For example, F(a, 8,7, 6;t,q) denotes the family of all unions U V; of an œ—flat Vi, a B—flat Vo, a y—flat V3 and a 6—flat V4 in PG(t,q) which
are mutually disjoint where 0 < œ < 8< + < 6< t
›#n) in Ư(k — 1,qg) where œị = (q” — 1) /(q — 1) for
In order to prove Theorem 1.1, we prepare the following propositions which play an important role in solving Problem 1.1 or 1.2 -
Proposition 1.1 (Hamada [4]) (1) If F € Zu(e,w;£,g) in the case £ > + 1, then F is a {ev1 + vy41,€¥0 + v,;t,q}—minihyper
(2 lf F € Fue, 1, Hạ; vo oBaits q) in the case h > 2 and t > pp_1 + wa +1, then F
is a {en + Š 'Ðu;-+_1; €U0 + > Vass tai
i=1 t=1 minihyper
Remark 1.1 It is known (cf Theorems 2.2 and 2.3 in Hamada and Tamari 20) that Fu(e,u1,42,° ,#n3t,q) # O if and only if either (a) kh = 1 or (b) A > 2 and
t 2 tu_1 + tụ + 1 where (€, 1, Ha, - su) € U(t,q)
Proposition 1.2 (Hamada [3]) If there exists a {ev + S +ben+
Š %u¿; ‡, g}—minihyper #' for some ordered set (E;/,1; H2; - + * y #) in U(t, q), then | FNA |>
Š ®ụ¿~1 for any (t — 2)—fat A in PG(¿,g) and | FANG |= ` 0„;~1 for some (ý — 2)— Rat ' in PG(t,q) Let H; (j =1,2,- 4 +1) beg+1 "hyperpianes in PG(t,q) which con- tain G Then FN 4; is a fas 721 + ` Đụ; 6j00 + ` p;-13t,q p—minihyper in Hj for
att
»q + 1 where the 6; are nonnegative integers such that }~ 6; =e j=l
j=1,2,
Remark 1.2, (1) For any (¢ ~ 2)~flat @ in PG(t,q), there are q +1 hyperplanes in
Trang 32(2) There exists an {f,m;t,q}—minihyper F such that F C 2 for some 0—flat Q in PG(t,q) if and only if there exists an {f,m;0,q}—minihyper where 2 < 6 < ¢ and 0 <Sm < ƒ < 1+
Proposition L3 (Hamada [4]) Let (€, #1, 42,°++,#) be an ordered set in U(t,q) such
that either (a) h = 1 and ị > 2, (Ø) h > 2, e = 0, mì = 1, we > 2 and? > py_1 + wp
or (y) h > 2, 2 2andt > pa-1 +a Let 6; (7 =1,2,+++,q+1) be nonnegative h h qt integers such that Sổ = e lÝ there exists a {ev1 + D> vy,41,€v0+ 3} 0ạụ,; ‡, g}—minihyper j= ist i=1 h
F such that (a2) | FONG |= 2 a for some (¢ 2)—flat G in PG(t,q) and (b)
FO; € Fu(6;, 41 — 1,da — 1, -;#y — 1;£,g) for any hyperplane H; (1 <j <q+1)
which contains Œ, then f' € ZU(£; #1; Ha; ° - - sp) t, 9)
Proposition 1.4 (Hamada [3]) Let t, q, e, k and (¡= 1,2, -,h) be any integers
such that ý > 2, g > 3, e = 0 or 1, 1 < h < ¿and Ì < tị < dạ < - < tụ < ‡
(1) In the case h = 1, Ƒ' is a {£0i + 0u+1,£9ọ + 9„; É, q}—minihyper if and only if # € Zu(e.m¡t,q) h h (2) In the case h > 2 andt > pp_-it+pyatl, Fisa {en + ¥ v4,41,€%0 + oaitsa}- os * + t1 =1 minihyper if and only if F € Fy(e, 1; t2; * ' *› tạ: É, q) , h h (3) In the case h > 2 and ý < ,_1-+„, there is no {en + > ụ;-+1; E90 + » muita b- minihyper = =t
Proposition 1.5 (Hamada [7]) Lett > 2,qg>3 andl] <yp<t
(1) In the case t < 2y, there is no (20,41, 2v,;¢, g}—minihyper
(2) In the case ¢ > 24 +1, F is a {2vy41,2vy;t,q}—minihyper if and only if F €
F(u, w5t, 9)
Definition 1.1 Let V be a #—flat in PG(t,q) where 2< 6 <4 A set S of m points in V is called an m—arc in V if no 8+1 points in S are linearly dependent where m > 8 +1 For convenience sake, a set S of @ points in V is also called an Ø—arc in W if @ points in S are linearly independent
Remark 13 Let ej: and ej2 (¡ = 0,1, - it — 1) be nonnegative integers such that ” (t- t O<en ten Sq-1 LF; isa 2 0/91£h 5 jot ]}—minihyper for j = 1,2 and i= t=] t-1 t—1 Fin F; = 0, thea ALU F, isa i= (eu + ex) ¥i41, 5 (en + °a)n¡l,4}—ninhgpe (cf eae =0 =1
(1), (2) and (3) in Defnition 11.2 and (4) in Defnition 1TT.2)
Definition 1.2 Let V and W be a ø—flat and a v—flat in PG(t,q), respectively, such
that V NW is an m—flat in PG(t,q) where O<m< psu <t Let V @ W denote the (u+v—m)-—flat in PG(t,¢q) which contain two flats V and W In the special case VNW = (Le„ ra = —1), V @ W denotes the ( + + 1)— at in PG(t,q) which contains V and W
Proposition 1.6 (Hamada [7]) Let e; (¢=1,2,-+-,#—1) be integers such that
either (a) 0 < e; $< g~-1 fori = 1,2, -,£ — 1 or (b) €1 = €2 = «++ = €)-1 = 0, €, = 4, 0S en41 <Sqg—1, 0< £¿_-¡ $< g—1 for some integer A in {1,2, -,¢ — 1}
Let H be a Ø—ñat in PG(t,g) where 2 <9 <1 If F is a set of points in H such that t-1 \FNG| > ¥ ex; for any (6 — 1)—flat G in H, then |FN HỊ 3> 3 gitat
t=1 i=1
Appendix II Preliminary results in the case q=3 In this appendix, let ¢ = 3 and v = (3Ì — 1) /(3 — 1) for any integer Í > 0 `
Proposition IL.1 (Hamada and Helleseth [16], [17]) Let t, «, 8 and + be integers such that either (a) 0 < œ = Ø < + < † and y ¿ œ+1 or (b)0 <Sœ< 8=+< ¿and y ¿ œ+1
(1) In the case t < Ø + +, there is no {Đa+i + 8+1 + Đrr1, Đạ + UB + By t, 3}—
minihyper
(2) In the case t > B+-y+1, F isa {va41 + vp4i + Yy4i, Ya + Ug + 0x¡ É, 3}— mỉnihyper
if and only if F € F(a,8,7;t,3)
Definition IL.1 (1) Let Ơ(0,1,1;Â,3) denote the family of all sets K in PG(t,3) such that K = V \ $ for some 2—flat V in PG(t,3) and some 4—arc S in V where t 2 2 (cf
Definition I.1 in Appendix 1)
(2) Let F(0,0,1;#,3) denote the family of all sets K in PG(t,3) such that K =
{(v1), (vo + 1), (240 + 1), (02), (Mì + 22),(Vo+ 211 + v2)} for some noncollinear points
(vo), (v1) and (v2) in PG(t,3)
Remark IL1 In this appendix, (v) and (2v) represent the same point in PG(¢, 3) for
any nonzero element v in the Galois field ŒF(4†?!)
Remark II.2 In (2) of Definition IJ.1, let & = vi, €1 = 2v9 + 21 and £2 = UỊ + 12
Then K = {(€0), (€1), (62), (2&0 + £1), (2€0 + €2), (261 + 2)} (cf (4) in Definition IT.2)
Trang 33Definition IL2 (1) Let F,(0,0,1,1;t,3) denote the family of all sets K in PG(t,3) such that K=V \ 5 for some 2~flat V in PG(t,3) and some 3—arc S in V where t >2
(2) Let 7z(0,0,1,1;‡,3) denote the family of all sets K in PG(t,3) such that K = (V \ S) U {P} for some 2—flat V in PG(t,3), some 4—arc S in V and some point P gV
where t > 3 ;
(3) Let F3(0, 0,1,1; 4,3) denote the family of all sets K in PG(t,3) such that K = LUK*
for some 1 flat L in PG(t, 3) and some minihyper K* in F(0,0,1;#,3) such that 1n K* = 0
where ý > 2 ,
(4) Let Ơ4(0,0,1,1;Â,3) denote the family of all sets K in PG(t,3) ? a 4,8, ,3) such that K =
{(£0), (£1), (€2), (€5), (260 + 1), (20 + £2); (20 + €s); (261 + &2), (261 + €5); (Zea + @a)}
for some four linearly independent points (£0), (£1), (€2) and (és) in PG(t,3) where ¢ > 3
Remark IL3 In Theorem 2.2 of Hamada, Helleseth and 'Ytrehus [18], let &) = 2x, £1 = c1m, 2 = cove and 3 = c3v3 Then the set of 10 points i › _ points in Theorem 2.2 can be
expressed as K in (4) of Definition JJ.2
Proposition 1L3 (Hamada [6] and Hamada, Helleseth and Ytrehus [18}) (1) In the case
t = 2, F is a {2v1 + 2v2,2vp + 2v1;2,3}— minihyper if and only if F € Fi(0, 0,1, 1; 2,3)
(2) In the case ¢ > 3, Ƒ' is a {20 + 2v2, 2v9 + 2v1;¢,3}—minihyper if and only if either
F € F(0,0,1,1;2,3) or F € F;(0,0,1,1;2,3) for some i in {1,2,3,4}
Appendix II Preliminary results in the case q=4
" In this appendix, let g = 4 and œị = (4Ì — 1) /(4 — 1) for any integer Ï > 0
Proposition HL1 (Hamada and Helleseth [13) Let t, a, Ø and + be integers such that either a) 0 < œ < 8< + < tor (b) 0 <Sœ= 8 <+< tand + ¿ œ+1
(1) In the case t < i
mg S A+, there is no {va41-+ 96‡1 † 9++1, 9a + 9g + 0y; ,4}—
(2) In the case t > Øđ++-+1 Fisa{vga1 + "
i 5 , 1+ 0441, Va + :#,4}—
if and only if F € 7(a,8,+;t, 4) { ‘a p+ yt, Ya + Ug + xi 4} minihyper
* Definition IL1 Let F(0,0,1;2,4) denote the family of all sets K in PG(t,4) such at K = {(wo), (w1), (we), (wo + wi), (wo + wa), (wi + we), (wo + wi + we)} for some
noncollinear points (wo), (w1) and (we) in PG(t,4) where ¢ > 2,
Proposition I1L2 (Hamada [6]) In the case t > 2, F is ,
có i (Ha > 2, a {201 + ve, 2up + 0t;t,4}—
minihyper if and only if either F € F(0,0,1;4,4) or F € F(0,0,1; 2,4) -
Definition THL2 (1) Let F1(0,0,1,1;2,4) denote the family of all sets K in PG(t,4) such that K = IgUL U{(cowy + wi + we), (c,wo + aw; + we), (cowo + da? + w2) } for some noncollinear points (wo), (w:) and (we) in PG(t,4) and some elements cg, c1 and c;
in {0,1,a,a7} where ¢ > 2, Lo = (wo) ®@ (wi), Li = (wo) ® (wa) and œ is a primitive element in GF(27) such that a? = ø + 1
(2) Let Fo(0,0,1,1;t,4) denote the family of all sets K in PG(t,4) such
that K = Lo U {(wa),(wi + wa), (wo + wi + wa), (wo + ows + we); (a?w9 + aw; +2),
(wo + a?w1 + we), (wo + a 0n + 0a) } for some noncollinear points (wo), (wi) and (we)
in PG(t,4) where t > 2 and Lo = (wo) ® (w1)
(3) Let Z2(0,0,1,1;,4) denote the family of all sets K in PG(t,4) such that K =
(Lo\{(w1)}) U (Li\{(wa)}) U (La\{(wi + w2)}U {(awi + wa), (aw + we)} for some
noncollinear points (wo), (wi) and (we) in PG(t,4) where ¢ > 2, Lo = (wo) ® (w1),
Ly = (wo) @ (we) and Le = (wo) © (wi + we)
(4) Let ¥4(0,0,1,1;2, 4) denote the family of all sets K in PG(t, 4) such that K = LUK* for some 1—flat L in PG(t,4) and some minihyper K* in F(0,0,1;2,4) such that LNK* = 0
where t > 3
Remark IIL1 If co = cox +c? in (1) of Definition IIJ.2, then K in (1) contains three
1—flats Lo, Ly and L* where L* = (cow + wi + +02) ® (ertmo + aw, + w2) (cf Definition
1.2 with respect to the notation @)
Proposition T1.3 (Hamada [5], [6}) In the case ¢ = 2, F is a {2m + 20a, 20 + 2o; 2,4}—minihyper if and only if # € F;(0,0,1,1;2,4) for some ¢ in
{3,3}
Proposition IIL4 (Hamada, Helleseth and Ytrehus [18}) In the case ý > 3, F
is a {2v + 2v2, 2v9 + 2v;;t,4}—minihyper if and only if either F € F(0,0,1,1;2,4) or
F ¢€ F;(0,0,1,1;t,4) for some i in {1,2,3, 4}
Appendix IV The correspondence between minihypers and codes meeting the Griesmer bound
Let S(k,q) be the set of all column vectors ¢, e'= (co,¢1, *,¢x-1), in W(k,q) such
that either c,_1 = l Of c = 1, e441 = ¡+2 = *** = cy_1 = 0, for some integer ? in
{0,1, -, — 2} where k > 3 and W(k,q) denotes a k—dimensional vector space consisting
of column vectors over GF(q) Then $(k,q) consists of (g* — 1) /(q — 1) nonzero vectors
in W(k,q) and there is no element o in GF(q) such that xa=0x1 for any two vectors x1
and xq in S(k,q) Hence the (q* — 1)/(q— 1) nonzero vectors in S(k,q) may be regarded
as (q* —1)/(q~1) points in PG(k — 1,4)
Proposition IV.1 (Hamada[4]) Let F be a set of f vectors in S(k,q) and let Ở be the subspace of V(n,q) generated by a k x n matrix (denoted by G) whose column vectors are all the vectors in 6(k,g) \ #' where ø = 0y — ƒ, 1 < ƒ < ty — Í and tị = (q — 1) /(q — 1) for any integer 7 > 0
Trang 34to [FN H,| + q*-! — f, ice,
to(z'@) = |Fn H;| + g*~! — ƒ, V.)
where w(x) and 2' denote the number of nonzero elements in the vector x and the transpose
of the vector z, respectively
(2) In the case k > 3 and 1 < đ< q*~Ì, Ở is an [n, k, d; q|—code meeting the Griesmer
bound if and only if F is a {u, — n,v_1 — a + d; k — 1, g}—minihyper
Remark IV.1 For any two k x n matrices G; and G2 whose column vectors are all the vectors in S(k,q) \ F, there exists an n x n permutation matrix P such that G2 = GP
Let C’ be an [n,k,d;q]—code meeting the Griesmer bound for some integers k,d and
q such that k > 3 and 1 < d<q Lt A= [A¿,aa, -;an] be a & xn generator
matrix of C Then there exists a unique vector b; in S(k,q) for each vector a; in A such that bị = 9;a; for some nonzero element o; in GF(q) This implies that B = AD for a nonsingular diagonal matrix D=diag(o1,02, -,on) where B = [b,,b,, -, by] Hence we
introduce an equivalence relation among [n, k, d; g]—codes as follows:
Definition IV.1 Two [n,k,d;q]—codes C, and C2 are said to be equivalent if there
exists a k x n matrix G2 of the code C2 such that Gz = GiPD (or G2 = G1 DP) for some
permutation matrix P and some nonsingular diagonal matrix D with entries from GF(q),
where G is a k x n generator matrix of Ci
From Proposition IV.1 and Definition IV.1, we have
Proposition IV.2 In the case k > 3 and 1 < d < qÈ—Ì, there is a one-to-one
correspondence between the set of all nonequivalent (n, k, d; g]—codes meeting the Griesmer bound and the set of all {u, — m:,0y_— — n: + đ; b — 1, q}—minihypers h Corollary IV.3 In the case k > 3 and d = g-! — (¢ + 5° gH), there is a one- t=1 to-one correspondence between the set of all nonequivalent [n, k, ; d; q|—codes meeting the h
Griesmer bound and the set of all {en + 3) Đụ;+1;£0g +- `
tz] i= > Đụy & — 1,4 }—minhypss h
where n = uy — (<n +} tat)
a 7 i=1
+
Corollary IV.4 In the case k > 3 and d = gt-! — 2 7", there is a one-to-one correspondence between the set of all nonequivalent (n, k, d; q|~codes meeting the Griesmer bound and th e set of all & Pacts Dy vk -1, q ¢—minihypers where n = vj -% DAH ` 3 ; References 1 A.A Bruen and R Silverman, Arcs and blocking sets II, Europ J Combin 8 (1987), 351-356 2 J.H Griesmer, A bound for error-correcting codes, IBM J Res Develop 4 (1960), 532-542
3 N Hamada, Characterization resp nonexistence of certain q—ary linear codes attaining
the Griesmer bound, Bull Osaka Women’s Univ 22 (1985), 1-47
4 N Hamada, Characterization of min-hypers in a finite projective geometry and its
applications to error-correcting codes, Bull Osaka Women’s Univ 24 (1987), 1-24
5 N Hamada, Characterization of {12,2;2,4}—min-hypers in a finite projective geometry PG(2,4), Bull Osaka Women’s Univ 24 (1987), 25-31
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2, 2;2,q}—min-hypers in a finite projective geometry, Graphs and Combin 5 (1989),
63-81 `
1 N Hamada, A characterization of some (n, k,d;q)—codes meeting the Griesmer bound using a minihyper in a finite projective geometry, to appear in Discrete Math., In Chapter 4 in “Combinatorial Aspect of Design Experiments”
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Griesmer bound for given integers k > 3, g > 5 and d = gh} — g® — q
(0<œ<<+<k~1or0<œ<<+<k—!1), In: First Sino-Franco Conference
on Combinatorics, Algorithms, and Coding Theory, Bull Inst Math Academia Sinica
16 (1988), 321-338
9, N Hamada and M Deza, Characterization of {2(q+1)+2, 2; ý, g}—min-hyper in PG(t, q) (t > 3, q > 5) and its applications to error-correcting codes, Discrete Math 71 (1988), 219-231
10 N Hamada and M Deza, A characterization of {vy41 + €,0,3¢,¢}—min-hyper and its applications to error-correcting codes and factorial designs, J Statist Plann Inference 22 (1989), 323-336
11 N Hamada and M Deza, A characterization of {2va+41 + 20941, 20a + 2ug;t,q}—
minhypers in PG(t,g) (¢ > 2, q > 5 and 0 < a < B < 2) and its applications to error-correcting codes, Discrete Math 91 (1991), xxx-xxx
12 N Hamada and T Helleseth, A characterization of some {3v2,3v1;t,q}- minihypers
and some {2v2 + vy41,201 + 0x;f,g}—minihypers (q = 3 or 4, 2 < 7 < ¢) and its applications to error-correcting codes, Bull Osaka Women’s Univ 27 (1990), 49-107 13, N Hamada and T Helleseth, A characterization of some minihypers in a finite projective
geometry PG(t,4), Europ J Combin 11 (1990) 541-548
14, N Hamada and T Helleseth, A characterization of some linear codes over GF(4) meeting the Griesmer bound, to appear in Mathematica Japonica 37 (1992)
15 N Hamada and T Helleseth, A characterization of some {3v,41,3v,;% — 1,q}— mini-
hypers and some (n, k, g*~! ~ 3g"; q)—codes (k > 3, ¢ > 5,1 <p < k — 1) meeting
the Griesmer bound, submitted for publication
16.N Hamada and T Helleseth, A characterization of some {20z+t + 0y+i;2Ua+
0ạ;k — 1,3}—minihypers and some (n,k,3*-! — 2.3% — 37;3)—codes (k > 3, 0 <
a <+7<k—1) meeting the Griesmer bound, to appear in Discrete Math
Trang 3517 N Hamada and T Helleseth, A characterization of some minihypers in PG(t,3) and some ternary codes meeting the Griesmer bound, submitted for publication
18 N Hamada, T Helleseth and @ Ytrehus, Characterization of {2(q + 1) + 2,2; t,q}— minihypers in PG((,g) ( > 3, q € {3,4}}), to appear in Discrete Math
19, N Hamada and F Tamari, On a geometrical method of construction of maximal t—linearly
independent sets, J Combin Theory 25 (A) (1978), 14-28
20 N Hamada and F Tamari, Construction of optimal linear codes using flats and spreads in a finite projective geometry, Europ J Combin 3 (1982), 129-141
21 T Helleseth, A characterization of codes meeting the Griesmer bound, Inform and Control 50 (1981), 128-159
22 R Hill, Caps and codes, Discrete Math 22 (1978), 111-137
23 G Solomon and J.J Stiffler, Algebraically punctured cyclic codes, Inform and Control 8 (1965), 170-179, 24 F Tamari, A note on the construction of optimal linear codes, J Statist Plann Inference 5 (1981), 405-411 25 F Tamari, On linear codes which attain the Solomon-Stiffler bound, Discrete Math 49 (1984), 179-191 Deligne-Lusztig Varieties and Group Codes - by Johan P Hansen Matematisk Institut, Aarhus Universitet, Ny Munkegade, 8000 Aarhus C, Denmark e-mail: matiphO@mi.aau.dk 1 Abstract
We construct algebraic geometric codes using the Deligne-Lusztig varieties {De-Lu] associated to a connected reductive algebraic group G defined over a finite field F,, with Frobenius map F The codes are obtained as geometric Goppa codes, that is linear error-correcting codes constructed from algebraic varieties [Gol] and [Go2] The finite group G* of Lie type acts as F,- rational automorphisms on the codes and they become modules over the group algebra Fy[G" ] Algebraic geometric codes with a group algebra structure induced from automorphisms of the underlying variety have been constructed and studied in [Hal] , [Ha2] , [Ha-St] and
[VỊ -
The Deligne-Lusztig varieties used in the construction of the codes have in some cases many F,-rational points, which ensures that the codes have a large word length In case G is of type 7A, the Deligne-Lusztig curve considered have 1+ @ points over F + In case Œ is a Suzuki group 7B, , respectively a Ree group 7G, » the Deligne-Lusztig curves considered have 1+ q’ , respectively 1+ 9° , points over F, In relation to their genera these numbers are maximal as determined by the ”explicit formulas” of Weil Tabel of Contents 1 Geometric Goppa codes as group codes 2 Deligne-Lusztig varieties 3 Groups of type 7A, 4 Suzuki groups 7B, 5 Ree groups 2G,
App Weil’s ’explicit formulas”
Trang 36Introduction
Many classical linear error-correcting codes can be realized as ideals in group- algebras or as modules over group-algebras S D Berman [Be] etablished that the Reed-Muller codes over F, are ideals in a group-algebra over an elementary abelian 2- group This was generalized by P Charpin [Chl], [Ch2] , [Ch3] and P Landrock and O Manz {La-Ma] , who showed that any Generalized Reed-Muller code is an ideal in a group-algebra over an elementary abelian p- group |
In [Hal] , [Ha2] and [Ha-St] H Stichtenoth and the author construct
algebraic geometric error-correcting codes with a group algebra structure In [V] S G Vladut shows that the asymptotically good codes on classical and Drinfeld modular curves [T-V-Z],[M-V] constructed by Yu I Manin, M A Tsfasman, Th Zink and himself can be realized as group codes
The general setup, which is treated in section 1, is to consider an algebraic variety X , defined over a finite field F,, with a group G of F q- tational automorphisms On the variety X we consider an F,-rational and G-invariant divisor D together with a G-stable set P,,P2, ,Pn of Fy-rational points on X, none in the support of D The associated Goppa code [Gol] and [Go2] is the image C= ¢(L(D)) © F} of the F¿-linear map:
$¿:L(D) — ~~ FF
f + (ÍCP\), (Pa)
The group G acts on the G-stable set P.,Pạ, ,Pa of F q-tational points giving Fy a F,[G]-module structure As the divisor D is G-stable, G acts on L(D) and it becomes a F,y[G]- module The F,-linear map:
, $:L(D) — t?
becomes a F,[G]-morphism, and the geometric Goppa code
_ C= ¢(L(D)) © Fy
a F,[G]- module
In [Hal] a series of geometric Goppa codes is obtained from the Klein quartic, codes which are ideals in the group-algebra F,[G], where G is a Frobenius group of order 21 In [Ha2] , [St2] and [Ti] series of geometric Goppa codes are constructed from Hermitian curves, among these there are codes that are ideals in the group-algebra F 2[G] , G being a non-abelian p-group of order q3, q = p", p prime In [Ha-St] another series of group codes are presented The codes are ideals in F,[5] , where S is a Sylow-2-subgroup of order * of the Suzuki-group of order đ(q—1)(+1) and ạ= 2?! The codes are geometric Goppa codes over Fy with good parameters
Section 2 introduces another series of varieties with large groups of automorphisms The varieties are Deligne-Lusztig varieties associated to a connected reductive algebraic groups G defined over a finite field F,, with Frobenius map F:G-—> G [De-Lu] Specifically, let G be a connected, reductive algebraic group G and let Xj be the F¿-scheme of all Borel subgroups of G with Frobenius morphism F: § gua Gc: For we W in the Weyl group
X(u) € X
is the subshceme of Xq of all Borel subgroups of G such that B and F(B) are in relative position w Let w = $,°+.-*Sn be a minimal expression for w Then
; X(#y .; 5n
is the space of sequences (By, ,8n) of Borel subgroups of Œ such that Bạ = FB, and B;_, and B; are in relative position e or s; The scheme
X(s,,+ »5n) is of dimension n and it is a compactification of fC) The F,- rational points of X(s,,.-.,5n) is X(e) and the finite group G* of Lie type acts as F,-rational automorphisms on X(5,,.-.,$n),X(w) and the F,-rational points X(e)
Section 3 treats the case where G is of type 7A, Then the finite group GF has order đŒ®(Œ—1 )(@Œ+ 1) For a simple reflection s€ W in the Weyl group, the curve X(s) is irreducible with genus
- (=9)
gi ng”
and it has 1+g'+2gg F ;-rational points Thỉs is the maximal number of F ¿ rational points a curve of thất genus can have according to the Weil bound
Section 4 treats the case where G is a Suzuki-group Then the finite group GF = 2Bo(q) » 9 27™*!, has order P(q-1)(P + 1) For a simple reflection s€ W in the Weyl group, the curve X(s) is irreducible with genus
,.#-
42
and it has 1+? F,-rational points This is the maximal number of F¿-rational points a curve of that genus can have according to Weil’s explicit bound discussed in the appendix In [Ha-St] plane models have been given and resulting codes have been - studied
Section 5 treats the case where G is a Ree group Then the finite group GF =7G,(q) , q@=3?"t!, has order @(q—-1)(¢? +1) ˆ, For a simple reflection s€ W in the Weyl group, the curve X(s) is irreducible with genus
;= ÊCE , Wao)
and it has 1+ q° F¿-rational points, which is the maximal number a curve of that genus can have according to Weil’s explicit bound discussed in the appendix
The appendix discusses Weil’s explicit formulas bounding the number of F,- rational points on the curves and thereby the length of the resulting geometric Goppa codes
I am grateful to J.P Serre who suggested me to study Deligne-Lusztig varieties in search of curves with large groups of automorphisms in relation to their
genera :
1 Geometric Goppa codes as group codes
1.1 Let X bea projective curve of genus g defined over a finite field Fg Let D be a Fg-rational and positive divisor and let L(D) denote the Fy: vectorspace of rational functions defined over Fg such that f=0 or div(f) >—D Finally let P,, P2, ,Pn be a set of F,y-rational points on X, none in the support of D The associated Goppa code [Gol] and [Go2] is the image C = @(L(D)) ¢ FY of the F,-linear map: -
(1.1.1) $:L(D) — FR
Trang 37Pa))-Theorem 1.2 (V D Goppa, cf [Gol],[Go2]) Assume 0 < degD <n The length n and the minimal distance d of the code C= ¢(L(D)) ¢ FF satisfies:
(1.2.1) d>n—degD
If X is smooth, the dimension k of the code C' satisfies:
(1.2.2) k= degD+i-—g for deg D > 2g—2
(1.2.3) k > degD+1—g for degD < 2g—2
In particular
(1.2.4) kid >i41]_-8,
1.3 From (1.2.4) it’s clear that geometric Goppa codes with good parameters are to be found on curves where g is small, that is on curves with a large number n of F,-rational points compared to the genus g As for the number WN of all F,- rational points on a curve, Weil’s bound asserts, that
(1.3.1) IN —(14a)| $ 2-94
With the “explicit formula” of Weil this general bound can in concrete cases be improved This technique is presented in the appendix and applied in the last 3 sections of this paper
1.4 Let G be a group of F,-rational automorphisms on the curve X The action of G on X induces an action on the divisors on X Assume that the divisor D is G-invariant and assume that the set P,,P2, ,Pn of F¿-rational points on X is G-stable The group G acts on the G-stable set P,,P, ,Pn of F,-rational points giving FG a F,[G]-module structure As the divisor D is
G-invariant, G acts on L(D) by f = fog" for feL(D) and gEG , and it
becomes a left F,[G]- module The F,-linear map: $:L(D) -— F?
of (1.1.1) becomes a F,[G]-morphism, and the geometric Goppa code
C= 4¢(L(D)) ¢ FF
a left F,[G]-module In case G acts freely and transitively on Pạ,Pạ, , Pa then the F,-vectorspace on the points can be identified with the group algebra F¿[G] and the geometric Goppa code C = ¢(L(D)) ¢ Ff becomes a left ideal in F,[G]
1.5 Let X be a projective curve of genus g defined over a finite field Fy of characteristic p Let G denote the automorphism group of X In case p=0 Hurwitz showed that G is finite and that
(1.5.1) |G| < 84(ø—1), p=0
In case p>O H L Schmidt showed that G is finite, but |G] is not bounded as above Stichtenoth [Stl] obtains (1.5.2) IG] < 1664 , p20 except in the case where X is defined by the affine equation 7 n n (1.5.2.1) frye ert! p">3 H.-W Henn [He] obtains (1.5.3) IG] < 89°, p20 excluding 4 cases In a footnote he asserts that this bound can be strengthen to (1.5.3.1) IG] < 3(29)*{29
excluding 2 more cases However in section 5 we will construct curves not on his list of excluded curves with more automorphisms than allowed by the proclaimed bound
(1.5.3.1)
2 Deligne-Lusztig varieties
Let & be an algebraically closed field of characteristic p
2.1 The basic properties of affine algebraic groups can be found in [Cal] and [Ca2] Here we recollect what is needed for our purpose An affine algebraic group G over k is an affine variety defined over k which is also a group such that the multiplication map Gx G-— G and the inversion map G— G are morphisms of varieties Every affine algebraic group is isomorphic to a closed subgroup of the general linear group Gl,(k) for some n An affine algebraic group is called simple if it has no non-trivial closed connected normal subgroup
2.1.1 The multiplicative group k* ~ GL,(k) is an algebraic group An algebraic group isomorphic to k*x xk* is called a torus A Borel subgroup B of a connected affine algebraic group G is a maximal connected solvable subgroup of G Any two Borel subgroups of G are conjugate in G A maximal torus lies in some Borel subgroup of G and two maximal tori in G are conjugate The group G has a maximal closed connected normal subgroup all of whose elements are unipotent This is the unipotent radical The group G is called reductive if the unipotent radical is trivial Let G be a connected, reductive algebraic groups G defined over the field & Let B be a Borel subgroup of G , let T be a maximal torus of G in B and let U be the unipotent radical of B.The Weyl group of G is the finite group W = N(T)/T where N(T) is the normalizer T in G
2.1.2 Let X = Hom(7,È*) be the character group of 7, that is the group of algebraic group homomorphisms from T to the multiplicative group &* and let likewise Y = Hom(&*,T) be the group of cocharacters of 7' X and Y are free abelian groups of the same finite rank Let xE€X and yEY By composition we obtain a morphism
etrse
so that yoy € Hom(k*,&*) and therefore of the form
(xo7)(A) = A™ lew
for some integer m
This gives a nondegenerate pairing XxY—¬ # (1) >> <Xx;1> where (xey)(A) = ASX!72 re The groups X and Y are in duality and there is a bijection between them X— Y aw av such that <a,a¥> = 2
2.1.3 Consider the finitely many minimal closed subgroups of U normalised by T Each of these are isomorphic to the additive group k.Anelement t€ T acts on any of these subgroups by conjugation, the corresponding automorhism k — k is multiplication by an element in &* Hence the action of T’ by conjugation gives an element in X = Hom(T,&*) for any minimal closed subgroups of U normalised by T These elements are called positive roots There is a unique Borel subgroup B- of G containing T such that BNB- = T Let U~ be the unipotent
radical of —, As before we consider the minimal closed subgroups of U~
Trang 38Hom(T,k*) for any of these groups These elements are called negative roots The action of the Weyl group W on T gives rise to actions of Won X and Y defined by
(2.1.3.1) (wx) t = x(w(2)), wEeW,xExX,teT ˆ
(wy) A = w(7(A)), wEeW,vyEeY,rAcK
For each root a there isan element wg € W such that Wg = 0 and trà = Ì
Let Ø C X be the set of roots Then { wa] a€6@} generates W and such that the action of the Weyl group on X and Y is determined by
(2.1.3.2) Wa(x) = x- <x,0%>a xe#x
y(Y) = y— <œy,y>œ ye Y
2.1.4 We have seen (2.1.3) that every connected reductive group G has a
root datum
(X,0,Y,0Ÿ)
associated to it, where X, Y are the character and cocharacter groups of a maximal torus of G and @ is the set of roots and 6” the set of coroots The root datum (X,6, Y,ø0Y) uniguely determines the connected reductive group G
2.1.5 Let G be a simple algebraic group Let gt denote the positive roots (2.1.3) A positive root is called simple if it can not be expressed as the sum of two positive roots Let {a,, ,a,} be the simple roots for G The Cartan integers
(2.1.5.1) Aj; =< a;,a,V>
takes on the value 2 if i=j and the values 0, —l, —2, —3 if if6j7 in such a way that the integers (2.1.5.2) ny = Aj; Aj takes on one of the values 0, 1, 2, 3 Let (2.1.5.3) 8; = Wy W i=1, ,] and let the order of $;8; be m,; Then : m-: W = <5, ,5] (5; =1 , (5;8;) 9 =1 for ifj> and * R35 = 0 > m = 2 (2.1.5.4) a; = 1 ° mis = 3 nz; == 3 Ne, > m = 6 a Q
The Dynkin diagram of is a graph with J nodes corresponding to the simple roots a; The nodes corresponding to different simple roots @; and a, are connected by n,; = A;; Aj; bounds An arrow is pointing from the node corresponding to a; to the node corresponding to @; if Aj; # —1
The Dynkin diagrams interesting for our purpose are the following:
Az o—~O
Bạ : asp
G, , ŒŒO
2.2 A connected reductive group G over & is isomorphic to a closed
subgroup af GLa(k) for some n (2.1) A map Fre? G-G is called a standard Frobenius map if Fr_¢ is the restriction of the morphism P¬_ P Fr lo = d6 Ũ on GL,(k) for some embedding dt ‘Gv ihto some GL,(k) A Frobenius map isa morphism , F:G—-4G
such that some power J“ is a standard Frobenius map The finite groups GF where G is a connected reductive group and F is a Frobenius map are called finite groups of Lie type The real number Q defined by -
(2.2.0.1) Q'` = p° where Ƒ° = Frie
will be of later importance
The fundamental theorem of Lang-Steinberg [L] and [Ste] asserts that the
map
(2.2.0.2) L: G— G, L(g)=g9'F(g)
is surjective This result has important consequences; in the following we recollect what is needed for our purpose
2.2.1 Let G be a connected reductive group, then G has a F- stable ‘Borel subgroup Namely let B be a fixed Borel subgroup, any other Borel subgroup is of the form gBg' for some g€G (2.1.1) The group F(B) is also a Borel subgroup, so there is an element g,@G such that go F(B)(g))"' = B From (2.2.0.2) we find a g€G such that L(g) = g9 }F(g) = gp Now gBg™ isa F-stable Borel subgroup as F(gBg™) = F(g) F(B) F(g!) =
(990) F(B)(990)* = 9B"
2.2.2 Let G be a connected reductive group, then any two F- stable Borel subgroups are conjugate by an element in G" For let By, B, be two F-stable Borel subgroups, then B, = gB)g7! for some g@G (2.2.1) As both groups are F- stable we have gi F(g) € Ng(5o)= Bo- By the theorem of Lang-Steinberg
applied to By there isa 6€ By such that L(6) = 6 'F(b) = g'F(q), that is
gò !1eGf and B, = (gb!) By( gb)
2.2.3 A Frobenius morphism induces a graph automorphism p of the Dynkin diagram (2.1.5) when the arrows are disregarded Let T be a maximally split torus The Frobenius morphism induces an action on the character and cocharacter groups of T: F:X — X " F(x)t = x(F(t)) xe* and F:¥ — Y
F(y)A = F(7(A)) +€Y
The action of F on the roots is related to the graph automorphism p of the Dynkin diagram Specifically F(p(a)) is a positive multiple of a each positive root a
The Dynkin diagrams with F-action interesting for our purpose are the following twisted groups where the F-action permutes the two roots:
7a, 7 aN
2B
Trang 392.2.4 If G has type 7A, , the real number Q (2.2.0.1) can take any value p’, e€N, e 0 These groups are defined with respect to a non-degenerate Hermitian form in 3 variables on F ẹ@ corresponding to the involution A+ A”
2.2.5 If G has type 7B, , then then characteristic p has to be 2 and the
real number Q (2.2.0.1) must satiesfy Q? = 27"*+!, neEN The finite groups
GF= 2B,(Q") are the Suzuki groups
2.2.6 If G has type 7G, , then then characteristic p has to be 3 and the real number Q (2.2.0.1) must satiesfy Q? = 37"+!, neEN The finite groups G"= ”G;(Q?) are the Ree groups
2.3 We now introduce the Deligne-Lusztig varieties associated to a connected reductive algebraic groups G defined over a finite field F,, with Frobenius map F:G — G [De-Lu] Specifically, let G be a connected, reductive algebraic group G and let Xp, be the F,y-scheme of all Borel subgroups of G with Frobenius morphism F: G7 The group G acts on X G by conjugation and for each Borel subgroup B of & the stabiliser of the corresponding point in X G is B, and their is a natural isomorphism G/B — Xqg.gr gBg'
2.3.1 The set of orbits of G in XAxX G can be identified with the Weyl group (2.2.1) For T a maximal torus and B a Borel subgroup containing it we have isomorphisms: W3 N(T)/T 3 B\ G/B G\(G/Bx G/B) 3 G\ (XgxXq) For wé W in the Weyl group the orbit of G in X cx*@ is denoted Gy O(w) an (2.3.1.1) X(u) € X
is the subshceme of X/„ of all Borel subgroups B of G such that (B,F(B)) € O() are in reÌative position tơ
2.3.2 The subscheme X(w) € Xv (2.3.1.1) is smooth of pure dimension
(2.3.2.1) , dim X(w) = I(w) = 2
where w = s,- +5, is a minimal expression for w as simple reflections [De-Lu,
1.3]
2.3.3 The subscheme X(u) € Xq (2.3.1.1) is GF - stable 2.3.4 Let w= 5,- *5, bea minimal expression for w Then
(2.3.4.1) Š(q; , 8n
is the space of sequences (Bo,- ,Bn) of Borel subgroups of G such that Ba = FB, and B;, and B; are in relative position e or s; The scheme ape) is of dimension n and it is a compactification of X(w) [De-Lu, 9.1]
23.5 The Fy-rational points of X(5,, ,8n) is X(e) and the finite
group G" of Lie type acts as F,-rational automorphisms on X (5,, ,5n) , X(w) and the F,-rational points X(e)
2.3.6 Let w= s,- +5n be a minimal expression for w as simple reflections Then X(s,, ,5n) is irreducible if and only if any simple reflection s€ W is in the F-orbit of some s; where i=1, ,n,[Lul,3.10 d]
2.3.7 The Euler characteristic of X(w) is according to [De-Lu, Theorem
7.1] determined by
o(G)~o(T) |G"
Stz(e) |T]
where T is a F-stable maximal torus contained in BEX(w), Stq is the Steinberg representation of Grand o(G) (resp o( T)) is the Frank of G (resp T) The order | Tử | is calculated by the formula
(2.3.6.1) x(X(u)) = (-—1)
(2.3/62) — jr] = | dety, (0716 #— 1)[,
cf [Ca2], where Y, = Hom(k*,7)) be the group of cocharacters of Tụ (2.2.1),
where Ty is a F-stable maximal torus contained in a F-stable Borel subgroup and the action of F and w on Y is described in (2.2.3) and (2.1.3)
2.4 In case sé W is a simple reflection we obtain from (2.3) that the Deligne-Lusztig variety
X(s) = X(s)UX(e)
is a curve with the group GF of Lie type acting as F,-rational automorphisms The F,-rational points on X(s) is X(¢) and the curve X(s) is irreducible if and only if any simple reflection s¢ W isin the F-orbit of s
2.4.1 The genus and Euler characteristic of X(s) = X(s)UX(e) is determined by
(2.4.1.1) 2-29 = x(X(w))+x(X(e)) _
which is calculated using (2.3.6.1) and the number of F,-rational points on X(s) is x(X(e)) , which is also calculated using (2.3.6.1)
3 Groups of type 7A, - Hermite curves There are two simple roots a,, œ; The corresponding Dynkin diagram has two nodes with 1 bound between them
The Cartan matrix (2.1.5.1) is ,
3.0.1 +2 —1
(8.0.1) —1 +2
The Frobenius morphism interchanges the two nodes If G has type 7A, , the real number Q (2.2.0.1) can take any value p°, e€N, e400 Let ¢= Q These groups are defined with respect to a non-degenerate Hermitian form 2‘ + y+ 2‘ = 0 on F 2 corresponding to the involution A > \! ‘The finite groups G* = 2Aa(#?) have order g°(q?—1)(q? +1)
The element wa, € W acts on the simple coroots according to (2.1.3.2) and the Cartan matrix (3.0.1)
Woy a) = aif - <ayon > oi! = yan V tạji(02 ) = ay — <a,,a7 >a, = af + a;
Trang 40(3.0.3) q
Stg is the Steinberg representation of 7A,(q’) and assumes the value St gle) =
q- ,
3.1 Now let T be a F-stable maximal torus contained in BE X(wg,) and le To be a F-stable maximal torus contained in B,€ X(e) Then according to (2.3.6.2) we obtain (3.1.1) jr} = dety ø g(0ã19# — 1)| = -1 1 0 1 1 0 , 0 1| °' “| tr 0 | o0 1 =te +1 (3.1.2) |TỂI = | dety gpleoF—1)] = 0 1 1 0 ; “1 1 o |} 0 1 =1
PROPOSITION 3.2 Let G be a connected reductive group of type 7A, over k=F 4 Let wa, € W be a simple reflection and let X (wa, ) be the corresponding Deligne-Lusztig variety Then X (wa, ) is an irreducible curve of genus
2 — #8 —f g= 5
with 1+q° points 3( g2 — 1) (G41 over F d Sets The finite group GF = 1A2(4?) has ° order fí{q )( + ) and it dcis as a group of F a7 rational automorphisms on X (we, )
The variety is a curve according to (2.3.2) and irreducible according to
(2.3.6) as the Frobenius interchanges the two simple roots From (2.3.7) we
determine the Euler characteristica using (3.1.1) and (3.1.2): (3.2.1) X(we,)) = (-1)9 697-7) ja? | KOR) SY Stg(e) [TF] 3 —_ 3 ~ Tản oe) = -—(#-1)(¢+1) (3.2.2) x(X(e)) = (-1)76 9) ~ 7%) GF | Sto(e) [To] - CŒ =1 +1) _ 3 g (¢—-1) tra and the genus using 2.4.1.1 (3.23) 2—2¢ = x(X(@)) + x(X(wa,)) = 1+j~ (Œ—1)(4+1) ° potas 2
The finite group GF = ?A2(g) order 9°(g?—1)(q° +1) and it acts as a group of F ¿ -rational automorphisms on X (a,) by (2.3.3)
PROPOSITION 3.3 The irreducible curve X(wq,) of genus
2
g= 54
with 1+q° points over F , has the mazimal number of rational points allowed by the Weil-bound The Zeta-fdaction of the curve is (1+ qt)! (—0(1—gÐ9) and the number Nm of F om rational points is determined by the formula 2(X,Fq)(t) = m Nm = 1+q"™-(i"+(-d)™) gq", meéeN
The claim follows from the general theory of Zeta-functions (see the appendix), as the formula for Nz is seen to be true by inspection
REMARK 3.4 It is possible to construct geometric Goppa codes over F ? such that
2 —
dimension + minimal distance > 1+ 4° — t=?
The codes are modules over the group-ring F pal? )}] These codes have been studied in detail in [ Ha2]
4 Groups of type 7B, — Suzuki groups There are two simple roots a, , a, The corresponding Dynkin diagram has two nodes with 2 bounds between them and an arrow from the node corresponding to a, to that of a, (2.2.4) The Cartan matrix (2.1.5.1) is
+2 —1 4.0.1
( ) —2 +2
The Frobenius morphism interchanges the two nodes If G has type 7B, , then