Existence of BSAs and cyclic BSAs of block size three

A k-sized balanced sampling plan avoiding adjacent units (or BSA(v,k,λ;α) in short) is a pair (X,B), where X is a v-set with a cyclic order (x1,x2,…,xv) and B is a collection of k-subsets of X called blocks, such that no pair of s-adjacent units (xi,xi+s) appears in any block where s=1,2,…,α, while any other pair of units appears in exactly λ blocks. Let Zv={0,1,…,v-1} denote the cyclic additive group of order v and (Zv,B) a BSA(v,k,λ;α). If Zv is an automorphism group of the BSA(v,k,λ;α), then (Zv,B) is said to be cyclic and denoted by CBSA(v,k,λ;α). In this paper, the necessary and sufficient conditions for the existence of cyclic BSA(v,3,λ;4) are established by using Langford sequences. Furthermore, by utilizing 3-IGDDs and a kind of auxiliary designs called BSA*(v,{2,3},λ;α), the necessary and sufficient conditions for the existence of a BSA(v,3,λ;4) are finally determined.