Blocking sets in handcuffed designs

In this paper we determine the spectrum of possible cardinalities of a blocking set in a H(v,3,A) and in a H(v,4,1). Moreover we construct, for each admissible ~9, a H(v,3,1) without blocking sets. Introduction A handcuffed design with parameters v, k, A, or for short an H(v,k,A), consists of a set of ordered k-subsets of a v-set, called handcuffed blocks. In a block (a., a z ' ... ,a!:) each element is said to be "handcuffed" to its neighbours, so that the block contains k-l handcuffed pairs (a l ,a). (a 2 ,a 3 ), ••• , (ak_l,a)::), the pairs being considered unordered. The elements in a block are distinct, so the handcuffed pairs are distinct as well. A collection of b handcuffed blocks forms a handcuffed design if (i) each element of the v-set appears in exactly r of the blocks (ii) each (unordered) pair of distinct elements of the v-set is handcuffed in exactly A of the blocks. It can be shown [9] that every element of V must occur in the interior (that is, not in the first or last position) of exactly u Australasian Journal of Combinatorics Z( 1993), pp.229-236 blocks. Further the following equalities can be shown: .\(v-l)(k-2) u2 (k-l) r.\k(v-l) 2(k-1) b.\v(v-l) 2(k-1) (1) It is well known [12J that a H(v,3,.\) exists if and only if val (mod 4) for A-l,3 (mod 4), v-1 (mod 2) for A-2 (mod 4), all w:3 for .\-0 (mod 4), and a H(v,4,1) exists if and only if val (mod 3). Let (V,B) be a handicuffed design H(v,k,A). A subset S of V is called a ~ ~ if, for every bEB, bnS~ and bn(V\S)~. Blocking sets have been investigated in projective spaces [1,2,13], in t-designs [3,4,6,7,8,10,llJ and in G-designs [5). Let BS(v,k,A)-{h: 3 H(v,k,.\) ~ a ~ oet S ~ ISI-h} . In this paper, we completely determine BS(v,3,A) and BS(v,4,1) for all admissible v. In pd.L·t...l~ular we prove t:hat ~ for VEO (mod 2) BS(v,3,.\){ v;l ~} , 2 for val (mod 2) and that -{ v-I BS(v,4,1) -32V+31 } for val (mod 3) . Moreover, in section 4, we exhibit, for every VEl (mod 4), w:9, a H(v,3,1) without blocking sets.