The spectrum of HSOLSSOM(hn) where h is even Original Research Article

In this paper, we describe a generalized product construction and a construction using Steiner pentagon systems to obtain holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOM). We investigate the existence of HSOLSSOM(hn) for even h. We first improve the known result for h = 2 and show that a HSOLSSOM(2n) exists for all n ⩾ 5 except possibly for n ∈ E = {8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 28, 32}. As a consequence, we then establish that for h  2 (mod 4) and h ⩾ 10 a HSOLSSOM(hn) exists for all n ⩾ 5 except possibly for n ϵ E. More conclusively, we show that for h  0 (mod 4), a HSOLSSOM(hn) exists if and only if n ⩾ 5. We are also able to apply our results to construct a unipotent SOLSSOM(62), the existence of which was previously unknown.