Existence of HSOLSSOMs of type 2nu12nu1

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2nu12nu1 (HSOLSSOM(2nu1)HSOLSSOM(2nu1)). For u⩾2u⩾2, necessary conditions for existence of such an HSOLSSOM are that u   must be even and n⩾3u/2+1n⩾3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n⩽9n⩽9 and n⩾3u/2+1n⩾3u/2+1 or (2) n⩾263n⩾263 and n⩾2(u-2)n⩾2(u-2). In this paper we show that in (1) the condition n⩽9n⩽9 can be extended to n⩽30n⩽30 and that in (2), the condition n⩾263n⩾263 can be improved to n⩾4n⩾4, except possibly for 19 pairs (n,u)(n,u), the largest of which is (53,28)(53,28).