Some new HSOLSSOMs of types hn and 1m u1

In this paper, two new direct construction methods are given for holey self‐orthogonal Latin squares with a symmetric orthogonal mate (HSOLSSOMs). Some new HSOLSSOMs using already known methods are also given. The known existence results for HSOLSSOMs of types 1^m u¹ and hⁿ are improved; for type 1^m u¹ there remain just four possible exceptions with u odd and 3 ≤ u ≤ 15; for type hⁿ, there are just two possible exceptions remaining, for (h, n) = (6, 12) and (6, 18). As a byproduct, the known existence results for three holey mutually orthogonal Latin squares (3 HMOLS) are also improved. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 435–444, 2001