Sets of mutually orthogonal latin squares with “like subsquares,” II

Let N(n) denote the maximum number of mutually orthogonal Latin squares of order n. In this article we increase the lower bound on N(n) for n = 36, 40, and 48 by exhibiting sets of Latin squares which establish N(36) ⩾ 5, N(40) ⩾ 5, and N(48) ⩾ 5. In each of these orthogonal sets the Latin squares are all transversal identifying and all contain like subsquares.