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