A few more BIBDs with k=8 or 9
نویسنده
چکیده
It has been shown that for k = 8 or 9, there exists a BIBD(v, k, 1) for all positive integers v == 1 or k (mod k(k-1)), with some possible exceptions. We show that such designs exist for 49 of these exceptional values. A balanced incomplete block design (BIBD) with parameters (v, k, 1) is a pair (X, A) where X is a v-set and A is a family of k-subsets (where 2 < k < v) called blocks, such that every pair of distinct points of X occurs in exactly one block of A. It is well known that (i) (v (ii) v(v 1) 0 (mod k-1), and 1) == 0 (mod k(k 1)) are necessary conditions for the existence of a BIBD(v, k, 1). For k = 8 or 9, these conditions reduce to the condition that v be congruent to 1 or k (mod k(k 1)). In previous papers [3,4] it has been shown that for any positive integer v == 1 or k (mod k(k-1)) there exists a BIBD(v,k, 1) with some possible exceptions. It is our purpose here to reduce this number of possible exceptions to 64 for k 8 and 122 for k 9. To construct the designs that eliminate these exceptions, we require other com-binatorial configurations. For the definitions of pairwise balanced design (PBD), transversal design (TD), group divisible design (GDD), PED-closed set, and the various composition constructions for PBDs, see [6]. We also adopt the notation of this reference. We adopt the notation B(K) = {v : a PBD(v,K, 1) exists} and Rk = {1' : a BIBD((k 1)1'+ l,k, 1) exists}. We denote by PBD(v,K U{q*}, 1) a PBD which has exactly one block of size q and all other block sizes in K. We use the notation v E B(K U {q*}) to indicate the existence of a PBD(v,K U {q*}). In what follows, we shall first investigate the case k 8, and then k = 9.
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ورودعنوان ژورنال:
- Australasian J. Combinatorics
دوره 10 شماره
صفحات -
تاریخ انتشار 1994