Existence of HSOLSSOMs of type 4nu1

This paper investigates the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 4u (briefly HSOLSSOM(4u)). For u > 0, the necessary conditions for existence of such an HSOLSSOM are (1) u must be even, and (2) u ≤ (4n−4)/3, and either (n, u) = (4, 4) or n ≥ 5. We show that these conditions are sufficient except possibly (1) for 36 cases with n ≤ 37, (2) for n ≥ 38, n odd and n < u ≤ (4n − 4)/3, and (3) for n ≥ 38, n even and n + 14 < u ≤ (4n − 4)/3. As an application of the main result, we are able to construct various types of new idempotent incomplete self-orthogonal Latin squares with a symmetric orthogonal mate (briefly ISOLSSOM).