Báo cáo toán học: "On optimal linear codes over F8" pptx

On optimal linear codes over F 8 Rie Kanazawa and Tatsuya Maruta ∗ Department of Mathematics and Information Sciences Osaka Prefecture Un iversity, Sakai, Osaka 599-8531, Japan maruta@mi.s.osakafu-u.ac.jp Submitted: Aug 20, 2010; Accepted: Jan 29, 2011; Published: Feb 14, 2011 Mathematics Subject Classification: 94B05, 94B27, 51E20, 05B25 Abstract Let n q (k, d) be the smallest integer n for which there exists an [n, k, d] q co de for given q, k, d. It is known that n 8 (4, d) =  3 i=0  d/8 i  for all d ≥ 833. As a continuation of Jones et al. [Electronic J. Combinatorics 13 (2006), #R43], we determine n 8 (4, d) for 117 values of d with 113 ≤ d ≤ 832 and give upper and lower bounds on n 8 (4, d) for other d using geometric methods and some extension theorems for linear codes. 1 Introduction We denote by F n q the vector space of n-tuples over F q , the field of q elements. A q-ary linear code C of length n and dimension k (an [n, k] q code) is a k-dimensional subspace of F n q . The Hamming distance d(x, y) between two vectors x, y ∈ F n q is the number of nonzero coordinate positions in x − y. The minimum distance of a linear code C is defined by d(C) = min{d(x, y) | x, y ∈ C, x = y} which is equal to the minimum weight of C defined by wt(C) = min{wt(x) | x ∈ C, x = 0}, where 0 is the all-0-vector and wt(x) = d(x, 0) is the weight of x. A q-ary linear code of length n, dimension k and minimum distance d is referred to as an [n, k, d] q code. The w eight distribution of C is the list of numbers A i which is the number of codewords of C with weight i. The weight distribution (w.d. for brevity) with (A 0 , A d , ) = (1, α, ) is also expressed as 0 1 d α · · · . A k × n matrix having as rows the vectors of a basis of C is called a generator ma trix of C. ∗ This research was partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 2054 0129. the electronic journal of combinatorics 18 (2011), #P34 1 A fundamental problem in coding theory is to find n q (k, d), t he minimum length n for which an [n, k, d] q code exists ([5]). An [n, k, d] q code is called optimal if n = n q (k, d). The Griesmer bound (see [11]) gives a lower bound on n q (k, d): n q (k, d) ≥ g q (k, d) := k−1  i=0  d q i  , where ⌈x⌉ denotes the smallest integer greater than or equal to x. An [n, k, d] q code C is called Griesmer if it attains the G r iesmer bound, i.e. n = g q (k, d). The values of n 8 (k, d) are determined for all d only for k ≤ 3, see [19]. See [3] and [13] for the known results on optimal [n, 4, d] 8 codes for d ≤ 112. It is known from Theorem 2 .1 2 of [5] t hat n 8 (4, d) = g 8 (4, d) for all d ≥ 833. So, we concentrate on finding optimal linear codes over F 8 of dimension 4 with minimum distance 113 ≤ d ≤ 832. Our results ar e summarized to the fo llowing theorem, see also Table 3. Theorem 1.1. (1) n 8 (4, d) = g 8 (4, d) for d ∈ {257, 258, 265-272, 385-392, 441-568, 577-580, 7 05-728, 769-784}. (2) n 8 (4, d) = g 8 (4, d)+1 for d ∈ {113-120, 286-288, 377, 378, 399, 400, 407, 408, 4 14-440, 702-704, 7 50-752, 757-768, 813-8 16, 820-832}. (3) n 8 (4, d) ≤ g 8 (4, d) + 1 for d ∈ {129-13 2, 193, 259-264, 273-285 , 321-328, 393-398, 401-406, 4 09-413, 569-576, 581-6 32, 641-701, 72 9-749, 753-756, 785-812, 817-819}. (4) g 8 (4, d) + 1 ≤ n 8 (4, d) ≤ g 8 (4, d) + 2 for d ∈ {121-128, 177, 185-192 , 225-227, 233-236, 241-246, 2 49-256, 289-316, 337-3 76, 379-384, 63 9, 640}. (5) n 8 (4, d) ≤ g 8 (4, d) + 2 for d ∈ {133-176, 194-206, 209-220, 329-336, 633-638}. (6) g 8 (4, d) + 1 ≤ n 8 (4, d) ≤ g 8 (4, d) + 3 for d ∈ {178-184, 221-224, 228-232, 237-240, 247, 248, 317-320}. (7) n 8 (4, d) ≤ g 8 (4, d) + 3 for d = 207, 208. We a lso give a new construction of a [g q (4, d), 4, d] q code for d = 2q 3 − 5q 2 + 3q, q ≥ 5 (Proposition 3.6) and prove n q (4, d) ≥ g q (4, d) + 1 for q 3 /2 − q 2 − q + 1 ≤ d ≤ q 3 /2 − q 2 for even q ≥ 4 (Theorem 5.23 ) . 2 Preliminary results In this section, we give the geometric method and some known results which will be used in the later sections. We denote by PG(r, q) the projective geometry of dimension r over F q . A j-flat is a projective subspace of dimension j in PG(r, q). 0-flats, 1-flats, 2-flats, (r − 2)-flats and (r − 1)-flats are called points, lines, planes, secundums and hyperplanes, respectively. We denote by F j the set of j-flats of PG(r, q) and denote by θ j the number of points in a j-flat, i.e. θ j = (q j+1 − 1)/(q − 1). We set θ j = 0 for j < 0. Let C be an [n, k, d] q code which does not have any coordinate position in which all the codewords have a zero entry. We always consider such codes throughout this paper. The columns of the electronic journal of combinatorics 18 (2011), #P34 2 a generator matrix of C can be considered as a multiset of n points in Σ = PG(k − 1, q) denoted also by C. We see linear codes from this geometrical point of view. An i-point is a point of Σ which has multiplicity i in C. Denote by γ 0 the maximum multiplicity of a point from Σ in C and let C i be the set of i-points in Σ, 0 ≤ i ≤ γ 0 . For any subset S of Σ we define the multiplicity of S with respect to C, denoted by m C (S), as m C (S) = γ 0  i=1 i·|S∩C i |, where |T | denotes the number of points in a set T in Σ. When the code is projective, i.e. when γ 0 = 1, the multiset C forms an n-set in Σ and the above m C (S) is equal to |C ∩ S|. A line l with t = m C (l) is called a t-line. A t-pl ane and so on a re defined similarly. Then we obtain the partition Σ =  γ 0 i=0 C i such that n = m C (Σ), n − d = max{m C (π) | π ∈ F k−2 }. Conversely such a partition Σ =  γ 0 i=0 C i as above gives an [n, k, d] q code in the natural manner. For an m-flat Π in Σ we define γ j (Π) = max{m C (∆) | ∆ ⊂ Π, ∆ ∈ F j }, 0 ≤ j ≤ m. We denote simply by γ j instead of γ j (Σ). It holds that γ k−2 = n − d, γ k−1 = n. Lemma 2.1. For two distinct t-flats δ 1 and δ 2 in a fixed (t+1)-flat ∆ in Σ, 1 ≤ t ≤ k−2, it holds that m C (δ 1 ) + m C (δ 2 ) ≥ m C (∆) − (q − 1)γ t + q·m C (δ 1 ∩ δ 2 ). Proof. Considering the t-flats in ∆ through δ 1 ∩ δ 2 , we have m C (∆) ≤ m C (δ 1 ) + m C (δ 2 ) − m C (δ 1 ∩ δ 2 ) + (γ t − m C (δ 1 ∩ δ 2 ))(q − 1). Setting t = k − 2, a = m C (δ 1 ), b = m C (δ 2 ), c = m C (δ 1 ∩ δ 2 ) in Lemma 2.1, we get a + b ≥ (q − 1)d − (q − 2)n + qc. (2.1) When C is Griesmer, γ j ’s are uniquely determined as follows. Lemma 2.2 ([18]). For a Griesmer [n, k, d] q code, it holds for 0 ≤ j ≤ k − 1 that γ j = j  u=0  d q k−1−u  . By Lemma 2.2, every Griesmer [n, k, d] q code is projective if d ≤ q k−1 . Denote by a i the number of hyperplanes Π of Σ with m C (Π) = i and by λ s the number of s-points in Σ. When γ 0 = 2, we have λ 0 + λ 1 + λ 2 = θ k−1 and λ 1 + 2λ 2 = n, hence λ 2 = λ 0 + n − θ k−1 . (2.2) the electronic journal of combinatorics 18 (2011), #P34 3 The list of a i ’s is called the spectrum of C. Note that a i = A n−i /(q − 1). We usually use τ j ’s for the spectrum of a hyperplane of Σ to distinguish from the spectrum of C. Simple counting arguments yield the following three equalities. n−d  i=0 a i = θ k−1 . (2.3) n−d  i=1 ia i = nθ k−2 . (2.4) n−d  i=2 i(i − 1)a i = n(n − 1)θ k−3 + q k−2 γ 0  s=2 s(s − 1)λ s . (2.5) (2.3) and (2.4) yield the following: n−d−1  i=0 (n − d − i)a i = nq k−1 − dθ k−1 . (2.6) Furthermore, when γ 0 ≤ 2, we get the following from ( 2.3)-(2.5): n−d−2  i=0  n − d − i 2  a i =  n − d 2  θ k−1 − n(n − d − 1)θ k−2 +  n 2  θ k−3 + q k−2 λ 2 . (2.7) Lemma 2.3 ([22]). Let Π be an i-hyperplane through a t-secundum δ. The n (1) t ≤ γ k−2 − n − i/q = (i + qγ k−2 − n)/q. (2) a i = 0 if an [i, k − 1, d 0 ] q code with d 0 ≥ i −  i + qγ k−2 − n q  does not exist, where ⌊x⌋ denotes the largest integer less than or equal to x. (3) γ k−3 (Π) =  i + qγ k−2 − n q  if an [i, k − 1, d 1 ] q code with d 1 ≥ i −  i + qγ k−2 − n q  + 1 does not exist. (4) Let c j be the number of j-hyperplanes through δ other than Π. Then the following equality holds:  j (γ k−2 − j)c j = i + qγ k−2 − n − qt. (2.8) (5) For a γ k−2 -hyperplane Π 0 with spectrum (τ 0 , · · · , τ γ k−3 ), τ t > 0 holds if i + qγ k−2 − n − qt < q. The code obtained by deleting the same coordinate from each codeword of C is called a punctured code of C. If there exists an [n + 1, k, d + 1] q code C ′ which gives C as a punctured code, C is called extenda b l e and C ′ is an extension of C. We use the following extension theorems in Sections 4 and 5. the electronic journal of combinatorics 18 (2011), #P34 4 Theorem 2.4 ([6], [7]). Let C be an [n, k, d] q code with gcd(d, q) = 1, k ≥ 3. Then C is extendable if A i = 0 for all i ≡ 0, d (mod q). Theorem 2.5 ([24]). Let C be an [n, k, d] q code with q ≥ 5, d ≡ −2 (mod q), k ≥ 3. Then C is extendable if A i = 0 for all i ≡ 0, −1, −2 (mod q). Theorem 2.6 ([21]). Let C be an [n, k, d] q code with gcd(d, q) = 1 and assume that  i≡n,n−d (mod q) a i < q k−2 . Then  i≡n,n−d (mod q) a i = 0 and C is extendable. An [n, k, d] q code is called m-divisible if all codewords have weights divisible by an integer m > 1. The following theorem gives a restriction on the weights of a Griesmer [n, k, d] 8 code with d ≡ 0 (mod 8). Lemma 2.7 ([23]). Let C be a Griesmer [n, k, d] 8 code. If 8 divides d, then C is 2-divisible. In the remainder of this section, we give some known results on n q (4, d). Theorem 2.8 ([17]). n q (4, d) = g q (4, d) for all q for q 3 − 2q 2 + 1 ≤ d ≤ q 3 − 2q 2 + q a nd for q 3 − q 2 − q + 1 ≤ d ≤ q 3 + q 2 − q. Theorem 2.9 ([17],[20]). For q ≥ 4, n q (4, d) = g q (4, d) + 1 for q 3 − q 2 − 2q + 1 ≤ d ≤ q 3 − q 2 − q and for 2q 3 − 3q 2 − q + 1 ≤ d ≤ 2q 3 − 3q 2 . Theorem 2.10 ([17]). n q (4, d) ≥ g q (4, d) + 1 for (1) 2q 2 − 2q + 1 ≤ d ≤ 2q 2 for q ≥ 4, (2) (ν − 1)q 2 − 3q + 1 ≤ d ≤ (ν − 1)q 2 for 4 ≤ ν < q with ν not dividing q, (3) 2q 3 − rq 2 − q + 1 ≤ d ≤ 2q 3 − rq 2 for q > r, r = 3, 4 and for q > 2 (r − 1), r ≥ 5. Corollary 2.11. (1) n 8 (4, d) = g 8 (4, d) for d ∈ {3 85-392, 441-568}, (2) n 8 (4, d) = g 8 (4, d) + 1 for d ∈ {433-440, 825-832}, (3) n 8 (4, d) ≥ g 8 (4, d) + 1 for d ∈ {113-128, 233-256, 297-320, 361-384, 761-768}. Theorem 2.12 ([20]). T here exist no [n, 4, n + s − q 2 ] q codes for q 3 − sθ 1 − q + 1 ≤ n ≤ q 3 − sθ 1 for s = 2, q ≥ 4 and for s = 3, q ≥ 7, q = 9. Corollary 2.13. n 8 (4, d) ≥ g 8 (4, d) + 1 for 417 ≤ d ≤ 432. the electronic journal of combinatorics 18 (2011), #P34 5 3 Upper bounds on n 8 (4, d) Recall that the existence of an [n, k, d] q code implies t he existence of an [n − 1, k, d − 1] q code. So, from (1) and (2) of Corollary 2.11, it suffices to prove the following proposition in o rder to give the upper bounds on n 8 (4, d) in Theorem 1.1. Proposition 3.1. ( 1) There exist [g 8 (4, d) + 2, 4, d] 8 codes for d ∈ {128 , 136, 144, 152, 160, 168, 176, 177, 192, 200, 206, 216, 220, 227, 236, 246, 296, 304, 312, 316, 336, 344, 352, 360, 368, 640}. (2) There exist [g 8 (4, d) + 1, 4, d] 8 codes for d ∈ {120, 132, 193, 280, 288, 328, 378, 400, 408, 416, 424, 432, 576, 584, 592, 600, 608, 616, 624, 632, 648, 656, 664, 672, 680, 688, 696, 704, 736, 744, 752, 760, 768, 792, 800, 808, 816, 824}. (3) T here exist [g 8 (4, d), 4, d] 8 codes for d ∈ {258, 272, 580, 712, 720, 728, 776, 784}. As a method to construct good codes, we first introduce the projective dual. Lemma 3.2 ([15]). (1 ) There exists a [39, 4, 32] 8 code with w.d . 0 1 32 1911 36 2184 . (2) T here exists a [121, 4, 104] 8 code with w.d . 0 1 104 3136 112 945 120 14 . Lemma 3.3 ([22]). Let C be an m-divisible [n, k, d] q code with q = p h , p prime, whose spectrum is (a n−d−(w−1)m , a n−d−(w−2)m , · · · , a n−d−m , a n−d ) = (α w−1 , α w−2 , · · · , α 1 , α 0 ), where m = p r for some 1 ≤ r < h(k − 2) satisfying λ 0 > 0. Then there exists a t- divisible [n ∗ , k, d ∗ ] q code C ∗ with t = p h(k−2)−r , n ∗ =  w−1 j=0 jα j = ntq − d m θ k−1 , d ∗ = n ∗ − nt + d m θ k−2 = ((n − d)q − n)t whose spectrum is (a n ∗ −d ∗ −γ 0 t , a n ∗ −d ∗ −(γ 0 −1)t , · · · , a n ∗ −d ∗ −t , a n ∗ −d ∗ ) = (λ γ 0 , λ γ 0 −1 , · · · , λ 1 , λ 0 ). C ∗ is called the projective dual of C. Applying Lemma 3.3 to the codes in Lemma 3.2, we obtain the following codes. Corollary 3.4. (1) There exists a [312, 4, 272] 8 code with w.d . 0 1 272 3822 288 273 . (2) T here exists a [139, 4, 120] 8 code with w.d . 0 1 120 3297 128 749 136 49 . An f-set F in PG(r, q) with m = min{|F ∩ π| | π ∈ F r−1 } is called an {f, m; r, q}- minihyper. When an [n, k, d] q code is projective (i.e. γ 0 = 1) , the set of 0-points C 0 forms a {θ k−1 − n, θ k−2 − (n − d); k − 1, q}-minihyper, and vice versa, see [4]. Lemma 3.5. (1) There exists a [θ 3 − xθ 1 , 4, q 3 − xq] q code for 0 ≤ x ≤ q 2 − 1. (2) T here exists a [2q 3 − xθ 1 , 4, 2q 3 − 2q 2 − xq] q code for 0 ≤ x ≤ q 2 . the electronic journal of combinatorics 18 (2011), #P34 6 Proof. (1) Take x skew lines of PG(3, q) as the corresponding minihyper. (2) Let δ 1 and δ 2 be planes meeting in a line l in PG(3, q) and take skew x lines l 1 , · · · , l x not intersecting l. Deleting δ 1 , δ 2 and the skew x lines from two copies of PG(3, q), that is, setting C 0 = (δ 1 ∩ δ 2 ) ∪ (∪ 2 i=1 ∪ 1≤j≤x (δ i ∩ l j )), C 1 = (δ 1 ∪ δ 2 ∪ l 1 ∪ · · · ∪ l x ) \ C 0 and C 2 = PG(3, q) \ (C 0 ∪ C 1 ), we get the partition of PG(3, q) giving a generator matrix of the desired code. We get [g 8 (4, d)+2, 4, d] 8 codes for d = 336, 344, 352, 360, 368 and [g 8 (4, d)+1, 4, d] 8 codes for d = 400, 408, 416, 424, 432 from (1) of Lemma 3.5. We also get [g 8 (4, d) + 1, 4, d] 8 codes d = 808, 816, 824 from (2) of Lemma 3.5. A ( q 2 + 2q + 1)-set H in PG(3, q) which is projectively equivalent to the set {P(x 0 , x 1 , x 2 , x 3 ) ∈ PG(3, q) | x 0 x 1 + x 2 x 3 = 0} is called a hyperbolic quadric in PG(3, q), see [9]. H contains a set of q +1 skew lines called a regulus. H consists of (q + 1) 2 points, which are all the points on a pair of reguli. Using this property, we give a new construction of a non-projective Griesmer code as follows, which yields a [833, 4, 728] 8 code. Table 1: Codes obtained by Lemmas 3.7 and 3.8. C 1 C 2 C Lemma [139, 4, 120] 8 [10, 3, 8] 8 [149, 4, 128] 8 3.7 [139, 4, 120] 8 [15, 3, 12] 8 [154, 4, 132] 8 3.7 [94, 4, 80] 8 [65, 4, 56] 8 [159, 4, 136] 8 3.8 [103, 4, 88] 8 [65, 4, 56] 8 [168, 4, 144] 8 3.8 [112, 4, 96] 8 [65, 4, 56] 8 [177, 4, 152] 8 3.8 [121, 4, 104] 8 [65, 4, 56] 8 [186, 4, 160] 8 3.8 [130, 4, 112] 8 [65, 4, 56] 8 [195, 4, 168] 8 3.8 [139, 4, 120] 8 [65, 4, 56] 8 [204, 4, 176] 8 3.8 [312, 4, 272] 8 [65, 4, 56] 8 [377, 4, 328] 8 3.8 [650, 4, 568] 8 [10, 3, 8] 8 [660, 4, 576] 8 3.7 [585, 4, 512] 8 [80, 4, 68] 8 [665, 4, 580] 8 3.8 [605, 4, 528] 8 [65, 4, 56] 8 [670, 4, 584] 8 3.8 [614, 4, 536] 8 [65, 4, 56] 8 [679, 4, 592] 8 3.8 [623, 4, 544] 8 [65, 4, 56] 8 [688, 4, 600] 8 3.8 [632, 4, 552] 8 [65, 4, 56] 8 [697, 4, 608] 8 3.8 [641, 4, 560] 8 [65, 4, 56] 8 [706, 4, 616] 8 3.8 [650, 4, 568] 8 [65, 4, 56] 8 [715, 4, 624] 8 3.8 [585, 4, 512] 8 [139, 4, 120] 8 [724, 4, 632] 8 3.8 [585, 4, 512] 8 [149, 4, 128] 8 [734, 4, 640] 8 3.8 [449, 4, 392] 8 [312, 4, 272] 8 [761, 4, 664] 8 3.8 [724, 4, 632] 8 [64, 3, 56] 8 [788, 4, 688] 8 3.7 [724, 4, 632] 8 [73, 3, 64] 8 [797, 4, 696] 8 3.7 [512, 4, 448] 8 [312, 4, 272] 8 [824, 4, 720] 8 3.8 [531, 4, 464] 8 [312, 4, 272] 8 [843, 4, 736] 8 3.8 [540, 4, 472] 8 [312, 4, 272] 8 [852, 4, 744] 8 3.8 [549, 4, 480] 8 [312, 4, 272] 8 [861, 4, 752] 8 3.8 [558, 4, 488] 8 [312, 4, 272] 8 [870, 4, 760] 8 3.8 [567, 4, 496] 8 [312, 4, 272] 8 [879, 4, 768] 8 3.8 [576, 4, 504] 8 [312, 4, 272] 8 [888, 4, 776] 8 3.8 [585, 4, 512] 8 [312, 4, 272] 8 [897, 4, 784] 8 3.8 [585, 4, 512] 8 [322, 4, 280] 8 [907, 4, 792] 8 3.8 [585, 4, 512] 8 [331, 4, 288] 8 [916, 4, 800] 8 3.8 the electronic journal of combinatorics 18 (2011), #P34 7 Proposition 3.6. There exists a [g q (4, d), 4, d] q code for d = 2q 3 − 5q 2 + 3q, q ≥ 5. Proof. Let H be a hyperbolic quadric in PG(3, q) and let l 1 and l 2 be two skew lines contained in H. We further take two skew lines l 3 and l 4 contained in H meeting l 1 and l 2 and four points P 1 , · · · , P 4 of H so that l 1 ∩ l 3 = P 1 , l 1 ∩ l 4 = P 2 , l 2 ∩ l 3 = P 3 , l 2 ∩ l 3 = P 4 . Let l 5 be the line P 1 , P 4  and let l 6 be the line P 2 , P 3 . We set C 0 = l 1 ∪ l 2 ∪ · · · ∪ l 6 , C 1 = (l 1 , l 3  ∪ l 1 , l 4  ∪ l 2 , l 3  ∪ l 2 , l 4  ∪ H) \ C 0 and C 2 = PG(3, q) \ (C 0 ∪ C 1 ), where l i , l j  stands for the plane containing l i and l j . Taking the points of C i as the columns of a g enerator matrix i times, we get the desired [2q 3 −3q 2 +1, 4, d] q code, which is Griesmer for q ≥ 5. The next two lemmas are well-known to construct good codes from old ones, see Table 1 for the resulting codes. Lemma 3.7 ([8]). Let C 1 be an [n 1 , k, d 1 ] q code an d C 2 be an [n 2 , k − 1, d 2 ] q code. Assume that C 1 has a codeword c 1 with wt(c 1 ) ≥ d 1 + d 2 . Then an [n 1 + n 2 , k, d 1 + d 2 ] q code C exists. Lemma 3.8 ([8]). If there exist an [n 1 , k, d 1 ] q code C 1 and an [n 2 , k, d 2 ] q code C 2 , then so does an [n 1 + n 2 , k, d 1 + d 2 ] q code C. We also constructed linear codes with parameters [206, 4, 177] 8 , [222, 4, 192] 8 , [224, 4, 193] 8 , [232, 4, 200] 8 , [239, 4, 206] 8 , [250, 4, 216] 8 , [255, 4, 220] 8 , [263, 4, 227] 8 , [273, 4, 236] 8 , [284, 4, 246] 8 , [297, 4, 258] 8 , [322, 4, 280] 8 , [331, 4, 288] 8 , [341, 4, 296] 8 , [350, 4, 304] 8 , [359, 4, 312] 8 , [364, 4, 316] 8 , [434, 4, 378] 8 , [743, 4, 648] 8 , [752, 4, 656] 8 , [770, 4, 672] 8 , [779, 4, 680] 8 , [806, 4, 704] 8 , [815, 4, 712] 8 , by puncturing or lengthening some good codes with the aid of a computer. 4 The spectra of some [n, 3, d] 8 codes As a preliminary for Section 5, we give the needed results on the spectra of [n, 3, d] 8 codes in this section. Table 2 can be obtained from the known results. Note that a Griesmer [64 − e, 3, 56 − e] 8 code corresponds to a {θ 1 + e, 1; 2, 8}-minihyper, which necessarily contains a line in PG(2, 8) if e ≤ 3 , see Chap. 13 in [10]. Lemma 4.1 ([3]). n 8 (3, d) = g 8 (3, d) + 1 f or d ∈ {13-16, 29-32, 37-40, 43-48} and n 8 (3, d) = g 8 (3, d) for other d . The following three lemmas give the characterization of [119, 3, 104] 8 , [118, 3, 103] 8 and [117, 3, 102] 8 codes for q = 8. Lemma 4.2. The spectrum of a [2q 2 −q −1, 3, 2q 2 −3q] q code with q ≥ 5 is (a q−1 , a 2q−1 ) = (3, θ 2 − 3). A (2q 2 − q − 1)-plan e is obtained from two copies of PG(2, q) with three non- concurrent lines del e ted. the electronic journal of combinatorics 18 (2011), #P34 8 Proof. Let C be an [n = 2q 2 − q − 1, 3, 2q 2 − 3q] q code with q ≥ 5. By Lemma 2.2, γ 0 = 2 and γ 1 = 2q − 1. Since (γ 1 − γ 0 )θ 1 + γ 0 = n, any line thro ugh a 2-point is a γ 1 -line. Hence a i = 0 for θ 1 + 1 ≤ i ≤ γ 1 − 1. Let l be a t-line containing a 1-point P. Considering the lines through P , we get n ≤ (γ 1 − 1)q + t, so t ≥ q − 1. Hence a i = 0 for 1 ≤ i ≤ q − 2. Suppose a 0 > 0. Considering the lines through a fixed point of a 0-line, we have n = qγ 1 + 0 − 1, which implies a γ 1 −1 > 0, a contradiction. Hence a 0 = 0. One can prove a θ 1 −1 = a θ 1 = 0 similarly for q ≥ 5 considering the lines though a fixed 1-po int. Hence a i = 0 for all i ∈ {q − 1, 2q − 1}. The spectrum of C follows from (2.6) and (2.3). Since 2(q − 1) + (q − 1)γ 1 = n, the three (q − 1)-lines are not concurrent. Table 2: The spectra of some [n, 3, d] 8 co des. parameters possible spectra reference [6, 3, 4] 8 (a 0 , a 1 , a 2 ) = (34, 24, 15) [13] [7, 3, 5] 8 (a 0 , a 1 , a 2 ) = (31, 21, 21) [13] [8, 3, 6] 8 (a 0 , a 1 , a 2 ) = (29, 16, 28) [13] [9, 3, 7] 8 (a 0 , a 1 , a 2 ) = (28, 9, 36) [13] [10, 3, 8] 8 (a 0 , a 2 ) = (28, 45) [13] [33, 3, 28] 8 (a 0 , a 3 , a 5 ) = (9, 16, 48) [2] (a 0 , a 1 , a 4 , a 5 ) = (4, 5, 28, 36) (a 0 , a 3 , a 4 , a 5 ) = (6, 10, 18, 39) [42, 3, 36] 8 (a 0 , a 4 , a 5 , a 6 ) = (4, 6, 24, 39) [1] (a 0 , a 3 , a 5 , a 6 ) = (3, 7, 21, 42) (a 0 , a 4 , a 6 ) = (3, 21, 49) (a 0 , a 2 , a 4 , a 6 ) = (2, 3, 18, 50) [60, 3, 52] 8 (a 4 , a 6 , a 8 ) = (3, 16, 54) [12] (a 0 , a 4 , a 7 , a 8 ) = (1, 1, 32, 39) (a 0 , a 5 , a 6 , a 7 , a 8 ) = (1, 1, 3, 27, 41) (a 0 , a 6 , a 7 , a 8 ) = (1, 6, 24, 42) [61, 3, 53] 8 (a 0 , a 6 , a 7 , a 8 ) = (1, 3, 21, 48) [10] (a 0 , a 5 , a 7 , a 8 ) = (1, 1, 24, 47) [62, 3, 54] 8 (a 0 , a 6 , a 7 , a 8 ) = (1, 1, 16, 55) [10] [63, 3, 55] 8 (a 0 , a 7 , a 8 ) = (1, 9, 63) [4] [64, 3, 56] 8 (a 0 , a 8 ) = (1, 72) [4] [70, 3, 61] 8 (a 6 , a 8 , a 9 ) = (1, 24, 48) [4] (a 7 , a 8 , a 9 ) = (3, 21, 49) [71, 3, 62] 8 (a 7 , a 8 , a 9 ) = (1, 16, 56) [4] [72, 3, 63] 8 (a 8 , a 9 ) = (9, 64) [4] [73, 3, 64] 8 a 9 = 73 [4] [92, 3, 80] 8 (a 0 , a 8 , a 12 ) = (1, 9, 63) [16] (a 4 , a 12 ) = (6, 67) (a 4 , a 8 , a 12 ) = (1, 10, 62) (a 8 , a 12 ) = (12, 61) [101, 3, 88] 8 (a 5 , a 13 ) = (5, 68) [16] (a 9 , a 13 ) = (10, 63) Lemma 4.3. A [2q 2 −q −2, 3, 2q 2 −3q −1] q code with q ≥ 7 is extendable and its spectrum is (a q−2 , a q−1 , a 2q−2 , a 2q−1 ) = (1, 2, q, q 2 − 2) or (a q−1 , a 2q−2 , a 2q−1 ) = (3, q + 1, q 2 − 3). Proof. Let C be an [n = 2q 2 − q − 2, 3, 2q 2 − 3q − 1] q code with q ≥ 7. By Lemma 2.2, γ 0 = 2 and γ 1 = 2q − 1. Since (γ 1 − γ 0 )θ 1 + γ 0 − 1 = n, the lines though a fixed 2-point is one (γ 1 − 1)-line and q γ 1 -lines, and a i = 0 for θ 1 + 1 ≤ i ≤ γ 1 − 2. Let l be a t-line containing a 1-point P. Considering the lines through P , we get n ≤ (γ 1 − 1)q + t, so q − 2 ≤ t. Hence a i = 0 for 1 ≤ i ≤ q − 3. the electronic journal of combinatorics 18 (2011), #P34 9 Suppose a θ 1 > 0. Let l be a θ 1 -line. Since n = (γ 1 − 1)q + θ 1 − 3, the lines (= l) through a fixed 1-point of l are three (γ 1 − 1)-lines and q − 3 γ 1 -lines, for γ 1 − 3 > θ 1 . Hence  i≡n,n−d (mod q) a i = a θ 1 = 1, which contradicts Theorem 2.6. Hence a θ 1 = 0. One can prove a 0 = a q = 0 similarly applying Theorem 2.6. Therefore a i = 0 for all i ∈ {q − 2, q − 1, 2q − 2, 2q − 1}. Applying Theorem 2.4, C is extendable. Hence C can be obtained from a [2q 2 − q − 1, 3, 2q 2 − 3q] q code C ′ by removing one coordinate. Let P be the point corresponding to the coordinate. There are two possible spectra as stated according to the cases that P is a 1-point or a 2-point, respectively. Lemma 4.4. A [2q 2 −q −3, 3, 2q 2 −3q −2] q code with q ≥ 7 is extendable and its spectrum is one of the followings: (a) (a q−3 , a q−1 , a 2q−2 , a 2q−1 ) = (1, 2, 2q, q 2 − q − 2), (b) (a q−2 , a q−1 , a 2q−3 , a 2q−2 , a 2q−1 ) = (2, 1, 1, 2q − 2, q 2 − q − 1), (c) (a q−2 , a q−1 , a 2q−3 , a 2q−2 , a 2q−1 ) = (1, 2, 1, 2q − 1, q 2 − q − 2), (d) (a q−1 , a 2q−3 , a 2q−2 , a 2q−1 ) = (3, 1, 2q, q 2 − q − 3), (e) (a q−1 , a 2q−3 , a 2q−1 ) = (3, q + 1, q 2 − 3). Proof. Let C be a [2q 2 −q −3, 3, 2q 2 −3q −2] q code with q ≥ 7. By Lemma 2.2, γ 0 = 2 and γ 1 = 2q − 1. Since (γ 1 − γ 0 )θ 1 + γ 0 − 2 = n, a j-line though a 2-point satisfies j ≥ γ 1 − 2, and a i = 0 for θ 1 +1 ≤ i ≤ γ 1 −3. Every t-line through a 1-point satisfies n ≤ (γ 1 −1)q +t, so q − 3 ≤ t. Hence a i = 0 for 1 ≤ i ≤ q − 4. Suppose a 0 > 0 and let l be a 0-line. Then λ 2 = q 2 − 2q − 4 + λ 0 ≥ q 2 − q − 3 from (2.2 ) , for λ 0 ≥ |l| = θ 1 . It follows from (2.1) that a 0 = 1 and a i = 0 for 1 ≤ i ≤ 2q − 5, for θ 1 ≤ 2q − 4. Calculating (2.6) − 2 · (2.7), we get a 2q−2 = 2q 3 − 3q 2 − 2q + 3 − 2qλ 2 ≥ 0. Hence λ 2 ≤ q 2 − 3q − 1, a contradiction. Thus a 0 = 0. Suppose a θ 1 > 0. Let l be a θ 1 -line. From (2.8) with i = θ 1 and t = 1, we have a θ 1 = 1 and a i = 0 for i ∈ {q + 1, 2q − 3, 2q − 2, 2q − 1}, for q − 2 > 4. Calculating (2.6) − 2 · ( 2 .7 ) , we get a 2q−2 = 2q 3 − 6q 2 + 8 − 2qλ 2 ≥ 0, whence λ 2 ≤ q 2 − 3q, contradicting (2.2). Hence a θ 1 = 0. Suppose a q > 0. Let l be a q-line and let Q be the 0-point of l. Since n = (γ 1 − 1)q + q − 3 and γ 1 −3 > θ 1 , every j-line (= l) through a fixed 1-point of l satisfies j ≥ γ 1 −2 = 2q −3. From (2.8) with i = q and t = 0, we have c q−3 + c q−2 + c q−1 + c q ≤ 1. Assume c q−3 = 1. Then a q−3 = a q = 1 and a i = 0 for i ∈ {q − 3, q, 2q − 3, 2q − 2, 2q − 1}. We have λ 2 ≥ q 2 − 2q from (2.2) since a (q − 3)- line contains four 0-points. Calculating (2.6) −2 · (2.7), we get a 2q−2 = 2q 3 − 5q 2 + 4q + 3 −2qλ 2 ≥ 0, whence λ 2 ≤ q 2 − 5q/2 + 2, a contradiction. One can get a contradiction similarly for the other cases c q−2 = 1; c q−1 = 1; c q = 1; c q−3 = c q−2 = c q−1 = c q = 0. Hence a q = 0. Thus a i = 0 for all i ∈ {q−3, q−2, q −1, 2q−3, 2q −2, 2q−1}. Applying Theorem 2.5, C is extendable. Hence by the previous lemmas, C can be obtained from a [2q 2 − q − 1, 3, 2q 2 − 3q] q code C ′ by removing two coordinates. Let P and Q be the point corresponding to the coordinates. There are five possible spectra (a)-(e) as stated, according to the cases (a) the electronic journal of combinatorics 18 (2011), #P34 10 [...]... q-ary linear codes of dimension four, Discrete Math 208/209 (1999) 427–435 [18] T Maruta, On the nonexistence of q-ary linear codes of dimension five, Des Codes Cryptogr 22 (2001) 165-177 [19] T Maruta, Griesmer bound for linear codes over finite fields, http://www.geocities.jp/mars39geo/griesmer.htm [20] T Maruta, I.N Landjev, A Rousseva, On the minimum size of some minihypers and related linear codes, ... 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[14] R Kanazawa, On the minimum length of linear codes of dimension 4, MSc Thesis, Osaka Prefecture University, 2011 the electronic journal of combinatorics 18 (2011), #P34 21 [15] A Kohnert, Best linear codes, http://www.algorithm.uni-bayreuth.de/en/research/ Coding Theory /Linear Codes BKW/ [16] I Landjev, L Storme, A study of (x(q + 1), x; 2, q)-minihypers, Des Codes Cryptogr 54 (2010) 135-147 [17] T... J.W.P Hirschfeld, Projective Geometries over Finite Fields 2nd ed., Clarendon Press, Oxford, 1998 [11] W.C Huffman and V Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003 [12] S Innamorati, F Zuanni, Minimum blocking configurations, Journal of Geometry 55 (1996) 86-98 [13] C Jones, A Matney, H Ward, Optimal four-dimensional codes over GF(8), Electronic J Combinatorics... is a 2-point, (d) P and Q are distinct 2-points, (e) P and Q are the same 2-points, respectively The following three lemmas give the characterization of [110, 3, 96]8, [109, 3, 95]8 and [108, 3, 94]8 codes for q = 8 The proofs are quite similar to the proofs of Lemmas 4.2-4.4 and hence we omit here, see [14] Lemma 4.5 The spectrum of a [2q 2 −2q−2, 3, 2q 2 −4q]q code with q ≥ 7 is (aq−2 , a2q−2 ) =... codeword with weight d cannot exist It follows from Corollaries 2.11 and 2.13 that in order to give the lower bounds on n8 (4, d) in Theorem 1.1, it suffices to prove the nonexistence of [g8 (4, d), 4, d]8 codes for d ∈ {177, 185, 221, 286, 399, 407, 414, 639, 702, 750, 757, 813, 820} Lemma 5.1 There exists no [938, 4, 820]8 code Proof Let C be a putative [938, 4, 820]8 code Let δ be an i-plane through a... Now, calculating 5 · (2.6) − (5.14) again gives 5a23 + 9a24 + 12a25 + 14a26 + 15a27 + 15a28 + 12a30 + 9a31 + 5a32 = −900, a contradiction The following theorem gives the nonexistence of [213, 4, 185]8 codes Theorem 5.23 There exists no [(q 3 − q 2 − 3q + 2)/2, 4, q 3/2 − q 2 − q + 1]q code for even q ≥ 4 Proof For a putative Griesmer [(q 3 − q 2 − 3q + 2)/2, 4, q 3/2 − q 2 − q + 1]q code, the spectrum . known results on optimal [n, 4, d] 8 codes for d ≤ 112. It is known from Theorem 2 .1 2 of [5] t hat n 8 (4, d) = g 8 (4, d) for all d ≥ 833. So, we concentrate on finding optimal linear codes over F 8 of. methods and some extension theorems for linear codes. 1 Introduction We denote by F n q the vector space of n-tuples over F q , the field of q elements. A q-ary linear code C of length n and dimension. On optimal linear codes over F 8 Rie Kanazawa and Tatsuya Maruta ∗ Department of Mathematics and Information