Existence of BSAs and cyclic BSAs of block size three (Q872110)

A balanced sampling plan avoiding adjacent units \(\text{BSA}(v,k,\lambda;\alpha)\) is a set of \(v\) cyclically ordered points together with a collection of \(k\)-subsets of points called blocks, so that two points that are at most \(\alpha\) apart in the cyclic order appear in no blocks, while those more than \(\alpha\) apart appear in precisely \(\lambda\) of the blocks. Using Langford sequences, a complete existence result is given for \(\text{BSA}(v,3,\lambda;4)\) that admit a cyclic automorphism, and this is in turn used to give a complete existence result for \(\text{BSA}(v,3,\lambda;4)\) in general.