On property D neofields and some problems concerning orthogonal latin squares

Abstract

This chapter focuses on property D neofields and some problems concerning orthogonal Latin squares. The concept of the property D neofield arose from the attempt to find an explanation for the nonexistence of pairs of orthogonal Latin squares of order 6. The intention was to formulate a standard method of constructing a pair of orthogonal squares of any sufficiently small order r distinct from 6. The reason for failure when r = 6 could then be observed. The standard method devised is essentially a modification of the Bose method for constructing a complete set of mutually orthogonal Latin squares from a field. A sufficient condition for the existence of a pair of orthogonal latin squares of order r is that an (r − 1) × (r − 1) matrix A*r should exist with the following properties: (1) the integers 0 to r − 1 appear at most once in each row and column, and the integer (r −1)−i never occurs in the ith row; (2) the main secondary diagonal consists entirely of (r − 1)'s; and (3) all other secondary diagonals comprise the elements 0, 1, …, (r − 2) written cyclically.