On Optimal (Non-Trojan) Semi-Latin Squares With Side N And Block Size N: Construction Procedure And Admissible Permutations

Abstract

There is a special family of the (n×n)/k semi-Latin squares called the Trojan squares which are optimal among semi-Latin squares of equivalent sizes. Unfortunately, Trojan squares do not exist for all k; for instance, there is no Trojan square for k n. However, the need usually arises for constructing optimal semi-Latin squares where no Trojan squares exist. Bailey [2] made a conjecture on optimal semi-Latin squares for k n and based on this conjecture, optimal non-Trojan semi-Latin squares are here constructed for k = n, considering the inherent Trojan squares for k < n. A lemma substantiating this conjecture for k = n is given and proved. In addition, the properties for the admissible permutation sets used in constructing these optimal squares are made evident based on the systematic-group-theoretic algorithm of Bailey and Chigbu [3]. Algorithms for identifying the admissible permutations as well as constructing the optimal non-Trojan (n × n)/k = n semi-Latin squares for odd n and n = 4 are given.