On the distances between Latin squares and the smallest defining set size

Abstract

We show that for each Latin square L of order n ≥ 2 , there exists a Latin square L’ ≠ L of order n such that L and L’ differ in at most 8√n̅ cells. Equivalently, each Latin square of order n contains a Latin trade of size at most 8√n̅ . We also show that the size of the smallest defining set in a Latin square is Ω(n³/²).