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

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 8n cells. Equivalently, each Latin square of order n contains a Latin trade of size at most 8n. We also show that the size of the smallest defining set in a Latin square is Ω(n3/2).