Spectrum Of Splitting3-designs With Block Size3×2

Splitting t-designs were first formulated by Huber in recent investigation of optimal (t-1)-fold secure splitting authentication codes. Lett v, u, k, λ, t be positive integers and t≤u and uk≤v. A splitting t-design, namely, a t-(v, u×,λ) splitting design is a pair (X,B) such that the following properties are satisfied:(1) X is a set of cardinality v (called points),(2) B is a collection of subsets of X (called blocks) each of cardinality uk, such that every B∈B is expressed as a disjoint union of u subblocks of size k: B=B1∪B2∪…∪Bu. and (3) for every t-subset {x1.x2.....xt} of X occurs together in exactly λ blocks B=B1∪B2∪…∪Bu such that x1∈Bi1, x2∈Bi2...., xt∈Bit (ij,j=1,2,...,t, different in any two). In paper [Liang M. and Du B., A new class of splitting3-designs. Des. Codes Cryptogr.,2011.60.283-290] and [Chee Y.M.. Zhang X. and Zhang H.. Infinite families of optimal splitting authentication codes secure against spoofing attacks of higher order. Adv. Math. Commun.,2011,5,59-68], they have been given that the necessary and sufficient condition for the existence of3-(v.3×2.1) splitting designs. In this paper, it will be completely determined that the spectrum problem of3-(v,3×2,λ) splitting designs, is that (1) λv(v-1)(v-2)≡0(mod48),(2) λ(v-1)(v-2)≡0(mod8),(3) λ(v-2)≡0(mod2) and (4)v≥6.