One problem of interest in the study of Latin squares is that of determining parameter pairs (n, r) for which there exists a maximal set of r mutually orthogonal Latin squares of order n. In this paper we prove the existence of maximal sets of (p − 3)/2 mutually orthogonal Latin squares of order p, p ⩾ 7 a prime congruent to 3 modulo 4, and of maximal sets of (p − 1)/2 mutually orthogonal Latin squares of order p, p > 7 a prime congruent to 1 modulo 4.
