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

Resumen
Let View the MathML source, k=1,2,3 be Latin squares on symbol set S containing v elements. The fine structure of three Latin squares L1,L2,L3 is defined to be (t,s) if s=v2-c1 and t=c2, where cl is the number of cells (i,j) such that View the MathML source for l=1,2. Denote by Fin*(v) the set of all integer pairs (t,s) for which there exist three pairwise distinct Latin squares of order v on the same set having fine structure (t,s). We determine the set Fin*(v) for any positive integer vgreater-or-equal, slanted8.