Some New Row-Complete Latin Squares

A row-complete Latin square (RCLS) of order n, L, is a Latin square in which for any two distinct symbols, X, y, there exist i, j, 1 <i < ~1, 1 <:j < n 1 such that L(i,j) k x and L(i,j I1) =y. Column complete Latin squares are defmed analogously, and a square is complete if it is both. row and column complete. RCLS are used in statistics for the design of sequential experiments. A finite group G is sequencilde if its elements can be arranged into a sequence a,, ff2 ,.-., a, in such a way that the partial products a,,a,a,;...,a,a, ... a, are all distinct. Gordon [2] has shown that the existence of a sequencible group of order n is a sufficient condition for the existence of a complete Latin square of order n. In the same paper Gordon proves the existence of a sequencible goup of order 2n for all ~1. Very little is known about RCLS of odd order. Sequencible groups have been four&for orders .21, 27, 39, 55 and 57 (see [l, 3, 51); however, it is known there are no sequencible groups of order 9 or 15. We offer the following construction. Let n =p . q andyconsider the group G = Z, X Z,. A. q X n array, A, with entries from Z, x Z, will be called a gemzruting set of order n if: