On Bhaskar Rao designs of block size four

We show that Bhaskar Rao designs of type BRD(v, b, r, 4, 6) exist for v = 0,1 (mod 5) and of type BRD (v, b, r, 4,12) exist for all v ≥ 4. Disciplines Physical Sciences and Mathematics Publication Details de Launey, W and Seberry, J, On Bhaskar Rao designs of block size four, Combinatorics and Applications, Proceedings of the Seminar on Combinatorics and its Applications in honour of Professor S.S.Shrikhande, (K.S.Vijayan and N.M.Singhi, (Eds.)), Indian Statistical Institute, Calcutta, 1984, 311-316. This conference paper is available at Research Online: http://ro.uow.edu.au/infopapers/1015 hOCiedirlg, rif the Seminar 011 COl1lbinatorics and Applications _ _ . . in honour of Prof. S. S. Shrikhande on iUs 65th birthda)' Indian Statistical InstjtuU Dl!l:tlmber 14-17, 1962, pp. 311_316. ON BHASKAR RAO DESIGNS OF BLOCK SIZE FOUR By WARWICK DE LAUNEY and JENNIFER SEBERRY Department! oj Applied Mathematic8, University of Sydney, N.S. W., Australia ABSTRAOT. We show that Bhaska.r Raodesigns of type BRD(v, b, T, 4, 6) exist for v"," 0, 1 (mod 5) a.nd of type BRD(v, b, T, 4,12) exist for all to;" 4. Let A, Band A+B be vxb matrices with entries 0,1. Then X = A-B is said to be a Bhaskar Roo de8ign with parameters BRD(v, b, r, k, 1\) when the following matrix equations are satisfied: xX' ~ r1 (1) (A+BIIA+B)' ~ (r-,\)I+M (2) J(A+B) ~ kJ. (3) X is a vxb matrix with entries 0, +1, -1 with row inner product zero and which, when the -1 elements are replaced by + 1, becomes the incidence matrix of a BIBD(V, b, r, k, 1\). ]'or example the following matrix is a BRD(6, 15, 10, 4, 6) : Example 1: There exists a BRD(6, 15, 10,4,6). Write-for-l ,I 1 1 1 1 I I 1 1 1 1 0 0 0 0 oj ! 1 1 1 0 0 0 0 1 1 I 1 o . 1 1 0 1 0 0 0 1 0 1 1 0 0 0 I 1 0 0 I 1 o 0 1 0 1 0 1 0 1 0 1 0 1 0 1 I -, o 1 0 _I These designs were first studied by Bhaskar Rao [I, 21 and may be used to obtain group divisible PBIBD with parameters v" = 2v . , /\1 = 0, b* = 26, A, ~ A/2, k" = k, 'm = 2, n = v. W AitWlCK DE LAUNEY AND JENNIFER 5EBlIIRBY The necessary conditions for the existence of a BRD(v, b, 1', k, It) are for k = 4, '\(v-I) ~ r(k-I) bk = vr ... (4) and other restrictions on the parameters have been found (see [3, 4]) when k *4. They have also been studied by Vyas [5] and Singh [6]. We use the notation BRD(v, k, A) for BRD(v, b, 1', k, It) as band r are dependent on v, k, It. In this paper we use the following known results (see [7]) restricted to the group Z2 : Theorem 1: Suppose there exists a BRD(k, j, AB } and (i) a BRD(v, k, 1t.4,) then there exisUl a BRD(v,,7, A.4,AD); (ii) a BIBD(v, k, A) then there exists a BRD(v, j, AAlJ). Or, as is obtained in a similar fashion: Corollary 2: Suppose there exists a pairwise balanced design B(K, II, v) where K = {k1 , ''', kl.>} and a BRD(ki, j, p.) for each let € K then there exists a BBD(v, j, '\1'». The next result is a slight improvement on the result of Lam and Seberry [7] where the existence of k-l mutually orthogonal latin squares was required. The result may be proved by adjusting the matrix in the proof of the original theorem. Theorem 3: Suppose there flxists a BRD(u, k, A) with a 8ubdeJJign on w points (the values w = 0 and 1 are allowed), a BRD(v, k, A) and k-2 mutually orthogonal Latin squarea then there exisUl a BRD(v(u-w)+w, k, A) with subdesigns on u, wand v points. Remark 4: In this paper we are interested in the case k = 4, so we only need a pair of orthogonal latin squares and hence u-w may take on any value other than 2 or 6. Hanani's theorem stated on p. 250 of Hall (8] states Theorem 5 (Hana.ni): La u 0, 1 (mod 5) then_ u e B(KA, 1) where Kl = {5, 6, 10, 11, 15, 16, 20, 35, 36, 40, 70, 71, 75, 76}. Remark: We now see that if u 0, 1 (mod 5) and there exists a BRD(k" 4, 6) for every k, e K~ then we have the existence of a BRD(u, 4, 6) using either Theorem 1 or Corollary 2 with Theorem 5. The m.ain theorem: First we establish: Theorem 6: Let p '# 5, odd, be a prime or prime po'wer. Then there is a BRD(p, !p(p-I), 2(P-1), 4, 6). ON B1lASKAR RAO DESIGNS OF BLOCK SIZE FOUR 313 Proof.Let 9 be a generator of the multiplicative group, G, of GF(p). Consider the initial sets, writing get for g« with the non-identity element of Za attached, D t = (0, g', 9;+1, gH2} where i = 0, 1, ... , t{P~3). The differences from D; are gi(g2~1), gj+l(g~ 1), gH1HI+'(g_1), g1:(:1'-1)+'(g1_1), 9il-1J+,+1(g_1)}. As i runs through 0,1, "" !(p-3) the totality-of elements from all E( including repetitions is 3 copies of G and 3 copies of G with the non-identity element attached, that is, t(P-3) _ 8 E, ~ 3G+3G, ,~ giving the result. Theorem 7: A BRD(v, 4, 6) exi818 for v = 0, 1 (mod 5). Proof: By the remark after Theorem 5 it is merely necessary to show the existence of a BRD(u, 4, 6) for u e K~. These are obtained as given in Table 1 by developing the indicated initial blocks. First we exhibit a BRD(8, " 6) , Example 2: There is a BRD(8, 28, 14, 4, 6). 1 100 111 111 111 111 000 000 000 000 1 1 0 0 I I o 0 [ -[ -[ -[ J-[ B B B 1 1 0 0 o 0 1 1 o 0 1 1 B 1 -1 -B B J-1 [ 1 00 1 1 o 0 I 1 o 0 0 1 1 1 00 0 00 0 o 0 0 -1 1 1 1-