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Some Pairwise Balanced Designs Malcolm Greig Greig Consulting 207-170 East Fifth St. North Vancouver BC, Canada, V7L 4L4 greig@sfu.ca Submitted: Oct. 6, 1998; Accepted: Oct. 23, 1999. Abstract A pairwise balanced design, B(K; v), is a block design on v points, with block sizes taken from K, and with every pair of points occurring in a unique block; for afixedK, B(K)isthesetofallv for which a B(K; v) exists. Aset,S, is a PBD-basis for the set, T ,ifT = B(S). Let N a(m) = {n : n ≡ a mod m},andN ≥m = {n : n ≥ m};withQ the corresponding restriction of N to prime powers. This paper addresses the existence of three PBD-basis sets. 1. It is shown that Q 1(8) is a basis for N 1(8) \ E, where E isasetof5definite and 117 possible exceptions. 2. We construct a 78 element basis for N 1(8) with, at most, 64 inessential ele- ments. 3. Bennett and Zhu have shown that Q ≥8 is a basis for N ≥8 \ E  , where E  is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. (Constructions for 35 of these possible exceptions are known.) Finally, we provide brief details of some improvements and corrections to the generating/exception sets published in The CRC Handbook of Combinatorial De- signs. Key words and phrases: BIBD, Pairwise Balanced Design AMS subject classifications: Primary 05B05. 1 the electronic journal of combinatorics 7 (2000), #R13 2 1 Introduction A pairwise balanced design, B(K; v), is a block design on v points, with block sizes taken from K, and with every pair of points occurring in a unique block; for a fixed K, B(K)isthesetofallv for which a B(K; v) exists. Aset,S, is a PBD-basis for the set, T ,ifT = B(S). Let N a(m) = {n : n ≡ a mod m}, and N ≥m = {n : n ≥ m};withQ the corresponding restriction of N to prime powers. This paper addresses the existence of three PBD-basis sets. The opening sections deal with useful known results and more general constructions. In Section 5, we give constructions that are particularly useful for the first of our two problems, then in Section 6, we show that Q 1(8) is a basis for N 1(8) \ E,whereE is a set of 5 definite and 117 possible exceptions; an application of this result is given in [5]. In Section 7, we construct a 78 element basis for N 1(8) with, at least, 14 essential elements. In Section 8, we look at what Q ≥8 is a basis for. Bennett and Zhu have already tackled the last problem, and give some applications [9, 13]; they have shown that Q ≥8 is a basis for N ≥8 \ E  ,whereE  is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. Results (without proofs) from this paper were incorporated into [12, Tables III.3.18–19]. Finally, in Section 9, we provide brief details of some improvements and corrections to the generating/exception sets published in The CRC Handbook of Combinatorial Designs, specifically [12, Tables III.3.17–19]. Although we sometimes give some non-existence results (phrased as definite excep- tions and essential elements above) we will not establish these here; they require tech- niques quite different from the ones we are using here. We will merely note that most of them can be established by some results of Wilson’s [41]: the integrality conditions, and the bound on the size of a flat; these conditions are also given in [12, Lemma 3.2, Theorem 3.1.2]. Remark 1.1 Wilson’s integrality conditions imply that if S ⊂ N 1(m) ,andifaB(S; v) exists, then it follows that v ∈ N 1(m) . 2 Notation The notation we use is fairly straightforward. A K-GDD is a group divisible design with block sizes taken from K, and with a group size vector of {G 1 ,G 2 , ,G n }; we will usually write the group size vector in exponential notation. We will only be concerned with the λ = 1 case; i.e., a pair of elements from different groups occurs in one block of the design, whilst a pair of elements from the same group appears in no block. A pairwise balanced block design is a GDD with a group size vector of 1 v ;itis denoted by B(K; v). If the block size is uniform, with K = {k} then this block design is referred to as a balanced incomplete block design (a BIBD), and denoted by B(k; v). If a B(k; v) exists, then the quantity r =(v − 1)/(k − 1) is called the replication number of the BIBD, and we say that r ∈ RN(k). the electronic journal of combinatorics 7 (2000), #R13 3 A transversal design, TD(k; t), is a {k}-GDD of group type t k . The set of all v for which a B(K; v) exists is denoted by B(K), and a similar definition is used for TD(k). The notation B(K ∪{k ∗ }; v) indicates there is exactly one block of size k in the design in addition to those of sizes from K (further blocks of size k can occur only if k ∈ K). Resolvable designs are designs that admit a partition of the block set into subsets of blocks that contain every point exactly once. These designs are denoted by the prefix R. If a RB(k; v) exists, then we say that r ∈ RRN(k), where RRN(k) ⊂ RN(k). Some notation that is specific to a section will be introduced in that section. One final notational convention we have adopted should be mentioned. We have labelled several lemmas by a one or two letter code. This is summarized in the key in the appendix, which gives the symbolic parameters used in the lemma where the code is introduced, but numeric values are substituted in the table of constructions. This device allows considerable compression of the table, yet retains all the information needed to verify the constructions. 3 Direct Constructions We start this section by summarizing some well-known results in the first lemma. Lemma 3.1 If q is a prime or prime power, then 1. q 2 ∈ RB(q), and 2. q 2 + q +1∈ B(q +1), and 3. q 3 +1∈ RB(q +1), and 4. q 3 + q 2 + q +1∈ RB(q +1). Proof : All these are standard results. The first two amount to the existence of the affine and projective geometries, AG(2,q)andPG(2,q). The resolvability of the unital design was shown by Bose (see [15, 26]), and the resolvability of the last design can be found in Lorimer [33]. Lemma 3.2 Embedded in PG(2, 2 N ) is a set of v(n)=2 n (2 n −1)2 N−n +2 n points whose incidences with the lines of the plane are all either 0 or 2 n . Furthermore, if m<n,the set of v(n) points contains a subset of v(m) points whose incidence with the lines of the plane is either 0 or 2 m . Proof : The first part of this result was shown by Denniston [19]. Denniston considers an irreducible second order curve, Q(x, y)=ax 2 + bxy + cy 2 over GF(2 N ) in the non- homogeneous coordinates (x, y) of the plane. Let G be any additive sub-group of order the electronic journal of combinatorics 7 (2000), #R13 4 2 n of the additive group GF (2 N ). Then the set of points satisfying Q(x, y) ⊂ G forms a Denniston arc, {v(n); {0, 2 n }},inPG(2, 2 N )withv(n)=2 N+n − 2 N +2 n ; note if we consider a sub-group H of order 2 m and take H ⊂ G, this yields a {v(m); {0, 2 m }}-arc whose points are a subset of the v(n) points, so that Denniston arcs can be taken to be nested. Corollary 3.3 If k is a power of 2, and n is a non-negative integer, then k(k−1)2 n +k ∈ RB(k). Proof : The resolvability was demonstrated by Seiden in the case n = 1, (see [39]), and her proof carries over to Denniston’s arcs. The device used is to consider a line that has zero incidence with the arc; the incidence of its points with the non-zero lines of the arc identifies the resolution classes. Corollary 3.4 The following designs exist: 1. B({9, 17 ∗ }; 120 + 17 = 137 = 8 ∗ 17 + 1); 2. B({9, 33 ∗ }; 232 + 33 = 265 = 8 ∗ 33 + 1); 3. B({9, 65 ∗ }; 456 + 65 = 521 = 8 ∗ 65 + 1); 4. B({17, 33 ∗ }; 496 + 33 = 529 = 8 ∗ 66 + 1); 5. B({9, 17}; 273 − 120 = 153 = 8 ∗ 19 + 1); 6. B({17, 33}; 1057 − 496 = 561 = 8 ∗ 70 + 1); 7. B({9, 17, 33 ∗ }; 496 − 232 + 33 = 297 = 8 ∗ 37 + 1). Proof : The first four examples are formed by adding a complete external line to the appropriate Denniston arc in planes of order 16, 32, 64 and 32. The next two designs are formed by taking the complements of the Denniston arcs used in parts 1. and 4. above. The final result comes from taking the difference of the two arcs of parts 2. and 4., in the plane of order 32, again adding a complete external line. Corollary 3.5 The following designs exist: 1. B({8, 9, 16, 17, 27 ∗ }; 291); 2. B({8, 9, 16, 17, 31 ∗ }; 295); the electronic journal of combinatorics 7 (2000), #R13 5 Proof : Again we use the 264 point difference of two Denniston arcs that we used in Corollary 3.4.7, and now we add an incomplete external line. If we have a projective plane, PG(2,q), then we may generate a TD(q+1; q)fromitby deleting a point, and using the lines through that point to generate groups. Since all the Desarguesian planes have ovals, we may consider how these oval points are distributed amongst the groups. Recall that if q is odd, then there are q + 1 points in the oval; (if D is the Singer difference set, then −D is such an oval). If q is even, with D =2D, then we may augment this set, −D,ofq + 1 points by the point {0} togetasetofq +2 hyperoval points, which we will also term an oval. Lemma 3.6 If q =2t +1is an odd prime power, then we may form a TD(q +1;q) such that the distribution of the oval points amongst the groups is of type 0 1 1 q or 0 t 1 2 2 t or 0 t+1 2 t+1 .Ifq =2t is an even prime power, then we may form a TD(q +1;q) such that the distribution of the oval points amongst the groups is of type 1 q+1 or 0 t 2 t+1 . Proof : To demonstrate this in the odd case, we either delete an oval point, or a non-oval tangent point, or a non-tangent point, and note the number of external lines, tangents and secants that the deleted point lay on. For the plane of even order, we delete an oval point, or a non-oval point; there are no tangents here. Lemma 3.7 {81, 585}⊂RB(9), and {73, 433, 577}⊂B(9). Proof : The first three values follow from Lemma 3.1 using q =8orq =9. Thevalues 433 and 577 result from constructions by Abel [1] and Buratti [17]. Lemma 3.8 If t ∈{1, 6, 7} or t is a power of 2, then 56t +8∈ RB(8). Proof : This follows from Lemma 3.1 using q =8orq = 7, or from Corollary 3.3 using k =8. Lemma 3.9 If q is a prime or prime power, and q ≡ 1mod8, and q ∈ {25, 89} and q<4096, then 7q +1∈ RB(8). Proof :Theresultsforq ∈{9, 17} are given above. The remaining constructions were given by Greig [20]. It can be shown that the restriction q<4096 is unneccessary, although we do not need that improvement here. Lemma 3.10 The {9}-RGDD of type 3 33 exists. Proof : This design was constructed by Mathon; see [28]. The remaining basic designs we need are transversals. We will briefly summarize some well known results. the electronic journal of combinatorics 7 (2000), #R13 6 Lemma 3.11 If q is a prime or prime power, then q ∈ TD(q +1). Lemma 3.12 If m ∈ TD(k +1), then m ∈ TD(k). Lemma 3.13 If {m, n}⊂TD(k), then mn ∈ TD(k). These last three lemmas can be combined to give a weaker version of MacNeish’s result [34]. Theorem 3.14 If m has no prime divisors less than k, then m ∈ TD(k +1). Lemma 3.15 TD(k +1)=RT D(k). Lemma 3.16 If t =3, then 8t ∈ TD(9), and if t =4, then 8t +1∈ TD(9). Proof : See [3]; for Wojtas’ TD(9; 48) see [18]. Unless otherwise noted, all the TDs we need are from [3]. The remaining designs we need are incomplete transversal designs (ITDs). Loosely speaking, an ITD, written as TD(k; m) − TD(k; a), is a design that could be completed to a TD(k; m) by adding the blocks of a TD(k; a) to the ITD; we do not actually need to have a TD(k; a)tohavean ITD. We have chosen to provide constructions for the ITDs we need, even though better values can sometimes be obtained from [4]. Lemma 3.17 If m ∈ TD(k), then the incomplete transversal TD(k; m) − TD(k;1) exists. Proof : Delete one block of the TD to get the ITD. The other two constructions of ITDs that we use are corollaries of Wilson’s basic construction [40]. Lemma 3.18 If m ∈ TD(10) and 0 ≤ n ≤ m, and {k,k +1}⊂TD(9), then the incomplete transversal TD(9; km + n) − TD(9; n) exists. Lemma 3.19 If m ∈ TD(n) and 8 ≤ n, then the incomplete transversal TD(9; 8m + n − 8) − TD(9; n) exists. 4 General Constructions Lemma 4.1 (Direct Product) If m ∈ TD(n), then mn ∈ B({m, n}). Proof : We just fill in the groups of the TD. the electronic journal of combinatorics 7 (2000), #R13 7 Theorem 4.2 Suppose we have a K-GDD with v points and a group size vector of {|G j | : j =1, ,g}, and, for the first g − 1 groups, we have a B(K ∪{w ∗ }; |G j | + w) with 0 ≤ w, and, for the last group we have a B(K; |G g | + w); then there exists a B(K; v + w). Proof : This is a standard result; we add w infinite points, and when we fill the first g − 1 groups, we align the w block with the infinite points, and refrain from using it for these groups. Note that the resulting design does not contain a w block, unless, possibly, when w ∈ K. Corollary 4.3 (Indirect Product) If (m − w) ∈ TD(n), and a B(K; m) containing a B(K; w) sub-design exists, (or a B(K; m) exists with w =1), then n(m − w)+w ∈ B(K ∪{n}). Theorem 4.4 (Wilson’s Fundamental Construction) Suppose there is a “master” K −GDD with g groups and a group size vector of {|G j | : j =1, ,g}, and a weighting that assigns a positive weight of w(x) to each point x.LetW (B i ) be the weight vector of the i-th block. If, for every block B i , we have a K  − GDD with a group size vector of W (B i ), then there exists a K  − GDD with a group size vector of {  x∈G j w(x):j = 1, ,g}. Proof : See [41]. Also note that the blocks in the final design have cardinalities in K  , rather than K. Theorem 4.5 If r ∈ RN(k), then a design with block size of k and a group vector of (k − 1) r exists. Proof : Deleting a point, and using its blocks as groups in a B(k;(k − 1)r +1), gives the GDD; note that this construction can be reversed; (see [25]). 5 Specific Constructions In this section, we start applying the results of the previous section to produce the tools for the second of the problems, that of constructing designs whose block sizes are in Q 1(8) ,whereQ 1(8) is the set of prime-powers congruent to 1 modulo 8. It will be convenient to define U 1(8) by: t ∈ U 1(8) ⇐⇒ 8t +1∈ B(Q 1(8) ). Lemma 5.1 There exist {9}-GDDs of type 8 9 and type 8 10 . Proof : Apply Theorem 4.5 to the 73 and 81 point designs of Lemma 3.7. the electronic journal of combinatorics 7 (2000), #R13 8 Lemma 5.2 If q =8t +1∈ RRN(8), then 56t +8∈ RB(8), and 64t +9∈ B({9,q ∗ }), and there exists a {9}-GDD of type 8 7t+1 (q − 1) 1 , and there exists a {9,q ∗ }-GDD of type 8 q . Proof : Complete the RBIBD with q =8t + 1 infinite points, and then delete either an infinite point, or a finite point, and use its lines to indicate the groups. Theorem 5.3 If there exists a K-GDD on v points, with group sizes contained in M, and K ⊂ RN(9) ∪  Q 1(8) \{25, 89}  , then M ⊂ U 1(8) implies v ∈ U 1(8) . Proof : Give each point a weight of 8, and apply Wilson’s fundamental construction. The needed components come from Theorem 4.5 for RN(9), and Lemma 3.9 via Lemma 5.2 for the other block sizes. We fill in the groups with a point at infinity to obtain the required result. Corollary 5.4 If there exists a {9, 10}-GDD on v points, with group sizes chosen from M, and M ⊂ U 1(8) , then v ∈ U 1(8) . Lemma 5.5 (Code T) If m ∈ TD(10) and 0 ≤ n ≤ m and {m, n}⊂U 1(8) , then 9m + n ∈ U 1(8) . Proof : Truncate one group of the transversal to size n, then use Corollary 5.4. Concentrating on the points in the last block yields the “last spike” and “block deletion” constructions given below. Lemma 5.6 (Code Ls) If m ∈ TD(n) and m ∈ U 1(8) , and n ≥ 9 and n ∈ Q 1(8) \ {25, 89}, then 9(m − 1) + n ∈ U 1(8) . Proof :Removem−1 points from n−9 groups of the transversal, retaining all the points of the last block, to give a {9, 10,n ∗ }-GDD of type m 9 1 n−9 . Now use Theorem 5.3. Lemma 5.7 (Code L) If m ∈ TD(n) and m ∈ Q 1(8) \{25, 89}, and n ≥ 9 and n ∈ U 1(8) , then 9(m − 1) + n ∈ U 1(8) . Proof : As in Lemma 5.7, we construct a {9, 10,n ∗ }-GDD of type m 9 1 n−9 .Nowfillin the groups and delete the big block to give a {9, 10,m}-GDD of type 1 9m−9 n 1 , and use Theorem 5.3. Lemma 5.8 (Code BD) If m ∈ TD(10) and 9 ≤ n ≤ 10 and {m, m − 1}⊂U 1(8) , then 10m − n ∈ U 1(8) . Proof :Removen points from one block of the transversal, then use Corollary 5.4. the electronic journal of combinatorics 7 (2000), #R13 9 Remark 5.9 Note that if n = 10, then m ∈ U 1(8) is not needed. Lemma 5.10 (Code R) If v ∈ RB(9) and 8n+1 ≤ v and n ∈ U 1(8) , then v+n ∈ U 1(8) . Proof : Add n new points to the blocks of n resolution sets, then use Corollary 5.4. Lemma 5.11 If m ∈ TD(k +1), then: 1. there exists a {k +1,m+1}-GDD of type k m m 1 ; 2. there exists a {k +1,m}-GDD of type k m (m − 1) 1 . Proof : Take the transversal, and fill in groups with 1 or 0 points at infinity and then delete a finite point and use its blocks as groups to give the result. Lemma 5.12 If k is a power of 2, then there exists a {k +1, 2k +1}-GDD on k(2k +3) points, and this design has a group vector of k 2k−1 (2k) 2 . Proof : In some ways, this lemma is a corollary of Lemma 3.2. The Seiden design can be embedded in PG(2, 2k). We now consider the non-Seiden points in PG(2, 2k), and delete one of these and use its blocks to define the groups. Lemma 5.13 There exist {9, 17}-GDDs with group vectors of 8 15 16 1 , 8 17 , 8 16 16 1 , 8 17 16 1 and 8 15 16 2 . Proof : This follows from Lemma 5.2 with t = 2 (twice), from Lemma 5.11 with m =16 and m = 17, and from Lemma 5.12. Mullin et al. [38, Lemma 6.12], used a construction similar to Lemma 5.14. We could also adapt their Lemma 6.20, but this yields nothing new here, as we have a rich set of constructions in the next six lemmas, covering essentially the same ranges. Five of these constructions of these lemmas were used previously by Greig and Abel [22]; Lemma 5.16 (code B) is new. Lemma 5.14 (Code A) If m ∈ TD(18) and 0 ≤ s ≤ t ≤ m and n = s + t, and {m, 2m, s, t}⊂U 1(8) , then 17m + n ∈ U 1(8) . Proof : Truncate two groups to sizes s and t, and give the points of these truncated groups a weight of 8. Give the points of one other group weight 16 and give all the other points weight 8. The needed component group vectors are 8 15 16 1 ,8 16 16 1 ,and8 17 16 1 , and are obtained from Lemma 5.13, so we may apply Wilson’s fundamental construction to give a design with a group vector of (8m) 15 (16m) 1 (8s) 1 (8t) 1 .Wefillthesegroupsin with the aid of a point at infinity. the electronic journal of combinatorics 7 (2000), #R13 10 Remark 5.15 Note that, provided we took s = 0, we could relax the condition m ∈ TD(18) to m ∈ TD(17). Lemma 5.16 (Code B) If m is a prime power with m ≥ 17, 0 ≤ t ≤ 17, n =2t ≤ m +1, and {m, m +2}⊂U 1(8) , then 17m + n ∈ U 1(8) . Proof : Using Lemma 3.6, we may construct a TD(17; m)withatleastt groups con- taining a pair of oval points; give these t pairs of points a weight of 16, and give all the other points a weight of 8. The needed component group vectors are 8 17 ,8 16 16 1 , and 8 15 16 2 , and are obtained from Lemma 5.13, so we may apply Wilson’s fundamental construction to give a design with a group vector of (8m) 17−t (8(m +2)) t .Wefillthese groups in with the aid of a point at infinity. Lemma 5.17 (Code C) If m ∈ TD(17) and 0 ≤ s ≤ t ≤ m and n =2m + s + t, and {m, m + s, m + t}⊂U 1(8) , then 15m + n ∈ U 1(8) . Proof : Give s points from one group a weight of 16, give t points from another group a weight of 16, and give all the other points a weight of 8. The needed component group vectors are 8 17 ,8 16 16 1 ,and8 15 16 2 , and are obtained from Lemma 5.13, so we may apply Wilson’s fundamental construction to give a design with a group vector of (8m) 15 (8m +8s) 1 (8m +8t) 1 . We fill these groups in with the aid of a point at infinity. Lemma 5.18 (Code D) If m ∈ TD(17), 0 ≤ s +t ≤ m, n = s +2t, and {m, 2m, n}⊂ U 1(8) , then 17m + n ∈ U 1(8) . Proof : Give s points from one group a weight of 8, give t points from the same group a weight of 16, give the points of another group a weight of 16, and give the points of all the other 15 groups a weight of 8. The needed component group vectors are 8 15 16 1 ,8 16 16 1 , and 8 15 16 2 , and are obtained from Lemma 5.13, so we may apply Wilson’s fundamental construction to give a design with a group vector of (8m) 15 (16m) 1 (8s +16t) 1 .Wefill these groups in with the aid of a point at infinity. Lemma 5.19 (Code E) If m ∈ TD(17) and 1 ≤ s ≤ m and 1 ≤ t ≤ m and 15(s−1)+ t ≤ m and n = m+t, and {m+1,m+s, m+t}⊂U 1(8) , then 16(m+1)+(s−1)+n ∈ U 1(8) . Proof : Give the points from the last block a weight of 16, give s − 1 other points from the first group a weight of 16, give t − 1 other points from the second group a weight of 16, and give all the other points a weight of 8. There are 15(s − 1) lines containing a pair of points with weight 16, one from group 1 and the other from groups 3 through 17; when assigning the extra t − 1 weight 16 points in the second group, we must avoid these lines; this is possible by hypothesis. The needed component group vectors are 8 17 , 8 16 16 1 ,8 15 16 2 ,and16 17 , and are obtained from Lemma 5.13, or from Lemma 3.8 with q = 16, so we may apply Wilson’s fundamental construction to give a design with a group vector of (8m +8) 15 (8m +8s) 1 (8m +8t) 1 . We fill these groups in with the aid of a point at infinity. [...]... 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Pairwise Balanced Designs Malcolm Greig Greig Consulting 207-170 East Fifth St. North Vancouver BC, Canada, V7L 4L4 greig@sfu.ca Submitted: Oct. 6, 1998; Accepted: Oct. 23, 1999. Abstract A pairwise. block. A pairwise balanced block design is a GDD with a group size vector of 1 v ;itis denoted by B(K; v). If the block size is uniform, with K = {k} then this block design is referred to as a balanced