Pointwise defining sets and trade cores

A block design D = (V, B) is a set V of v elements together with a set B of b subsets of V called blocks, each containing exactly k elements, such that each element of V occurs in precisely r blocks, for some positive integers rand k. D is called a t-design if every t-subset of V occurs in exactly At blocks, for some positive integer At. Such a design D is described as a t-( v, k, At) design. A t-( v, k, At) defining set has previously been defined as a set of blocks which is a subset of a unique t-(v, k, At) design. A defining set is now more broadly defined to be a set of full and/or partial blocks which is contained in a unique t-( v, k, At) design. It is a pointwise defining set if partial blocks are present. If only full blocks are present, it may be considered as either a pointwise or a blockwise defining set. The results presented here lead to useful tools for finding both pointwise defining sets of designs, and the relevant generalization of trades. Some examples are given to illustrate this. 1 Definitions and Well Known Results Here a summary is presented of results relevant to finding pointwise defining sets of designs. Many details and proofs regarding blockwise defining sets of designs can be found in K Gray [8], [9], [10]. Summaries of theoretical and practical results on this topic are given in Street (20], [21]. Previous papers on defining sets dealt with blockwise defining sets only. The definitions given here have been broadened so that they are relevant to both blockwise and pointwise defining sets. Australasian Journal of Combinatorics 16(1997), pp.51-76 A combinatorial design is a finite set V and a collection B of subsets of V called blocks, and is denoted (V, H). If at least one of the blocks in B is a proper subset of then the design is called It is said to be a block design if there exist positive integers k and r such that each block contains precisely k elements of V, and each element of V occurs in precisely r blocks. A block design is said to be a t-design if there is a positive constant At such that every subset of t elements of V appears in precisely At blocks of H. For ease of notation in describing t-designs we refer to a set of m elements as an m-set, and in writing blocks we drop the set notation. Example 1.1 The set of blocks {123, 134,256, 456} on the set V = {1, ... , 6} forms a block design with v 6, r = 2, b = 4 and k = 3. It is not a 2-design since not all pairs occur the same number of times; for example, 12 occurs once and 56 occurs twice. If V = {1, ... , 7} and Bl = {124, 235, 346, 457, 561, 672, 713}, then Fl = (V, HI) is a t-design, where t = 2, v = 7 = b, k = 3 = r and At 1. A t-design based on v elements with block size k is denoted as a t-(v, k, At) design. When t 2 it is called a balanced design, denoted (v, k, A), and in this case v :::; b. When equality holds, that is, when v = b, the design is said to be symmetric. A design with the property that any two blocks intersect in a constant number of elements is said to be linked. Any design which is both balanced and symmetric is also linked, with any two blocks intersecting in A elements. A t-design with no repeated blocks is said to be simple. The designs used as examples throughout this paper are simple, though the results given apply generally unless otherwise specified. Theorem 1.2 The following relationships between the parameters of a design must hold: (i) for any block design, vr = bk; and (ii) for any t-(V,k,At) design, At(~) = b(;). o Remark 1.3 Clearly any t-design is also an s-design for 0 :S s < t. Using the above equation to express both As and At, and then eliminating b from the equations obtained, gives A (v-s) t t-s As = (k-S) . t-s This formula must hold for all 0 :S s < t. Note that Al = rand AO = b. Given the parameters t, v, k and At, these equations can be used to calculate values for b, r and As for all s < t. Clearly, for a t-design to exist, each parameter calculated using these formulae must have an integer value.