The Nonexistence of Linked Block Designs with Latin Square Association Schemes

Abstract

The listing of partially balanced linked block designs by Roy and Laha (1957) contains no Latin square designs. The listing of designs with Latin square association schemes by Clatworthy (1956), which includes those given by Bose, Clatworthy and Shrikhande (1954) and by Bose and Shimamoto (1952), and the later listings by Chang and Liu (1964) and by Clatworthy (1967) contain no linked block designs. The question then arises whether any linked block designs exist having the Latin square association scheme. In this note a partial answer to the question is given. It is shown that there do not exist any linked block designs which are partially balanced with two associate classes and have the $L_i$ association scheme for $i = 2,3$ or 4.