Further results on smallest defining sets of well known designs

A set of blocks which can be a subset of only one t-( v, k, At) design has been termed a defining set of that design. In an earlier paper the author examined the smallest such sets of blocks for certain designs; that work is continued here for further designs. Improved lower bounds for the cardinality of a defining set are given for the affine and projective planes of order n. 1. Defining Sets: definitions and basic results Any design we consider is a collection of b blocks (k-subsets) chosen from a set of v elements. The term block design refers to a collection of blocks chosen in such a way that every element belongs to exactly r blocks. If k < v we say the block design is incomplete. If every subset of t elements belongs to exactly At blocks for some constant At, we call the design a t-design and indicate its param eters by t-(v, k, Ad. When t 2 we say the design is balanced. In this paper, only 2designs are considered so we will abbreviate 2-(v,k,At) to (V,k,A). In a previous paper (4] the author introduced the term defining set to ref er to a set of blocks which can be a subset of only one t-( v, k, At) design, denotin g the defining set by d( t-( v, k, At)). For example, the set of blocks R = {123, 145, 167} can he completed t o a (7,3,1) design in two distinct ways: by adjoining either T1 = {246, 257, 347, 356} or T2 = {247, 256, 346, 357}. Hence R is not a defining set of either design. But the set of blocks Q {123, 145, 246} can be completed to a (7,3,1) design onlyby adjoining the blocks {167, 257, 347, 356}. Hence Q is a defining set of that design . A minimal defining set, denoted by dm(t-( v, k, At)), is a defining set no pr oper subset of which is a defining set. A smallest defining set, denoted by ds(t-(v, k, At)), is a defining set such that no other defining set has smaller cardinali ty. Clearly, every t-( v, k, At) design has a defining set (the whole design) and hence a smallest defining set. A d( t-( v, k, At)) defining set consisting of blocks of a particula r t-(v, k,'x) design D is abbreviated to dD. The term trade is used to refer to two distinct collections of the same numb er of k-sets which contain precisely the same pairs (see Billington [1] and Gray [3J); fo r Australasian Journal of Combinatorics 1(1990), pp. 91-100 example, the collections T1 and T2 given above. Such collections are also known as mutually balanced (Rodger [5]). Every permutation on the elements of V induces a mapping from a k-set to a k-set. An automorphism of a set of blocks X is a permutation on the elements which takes every block of X to a block of X. Let Aut(X) denote the group of all the automorphisms of X. In Gray [4], the following results were established for incomplete block designs; they are now given without proof. LEMMA 1.1 Every defining set of a t-( v, k, Ad design D contains a block of every possible trade TeD. LEMMA 1 Suppose S is a particular defining set of a (v, k, A) design D and p E Aut(D). Then p(S) is also a defining set of D and Aut(S) is a subgroup of Aut(D). LEMMA 1.3. No automorphism of a 2-(v, k, 1) design, with k single trGms:po'slt1011. 2, consists of a LEMMA 1.4. Any d(2-( v, k, 1)) defining set S, for k occurring in its blocks. has at least (v-I) elements LEMMA 1.5. Suppose each of the elements i and j appears only once in a d( 2-( v, k, 1)) defining set S, where k > 2. Then i and j cannot appear in the same block of S. THEOREM 1.6. For every k,l) D,withk 2,