The spectrum of \(\text{HSOLSSOM}(h^ n)\) where \(h\) is even
An \(\text{HSOLSSOM}(h^n)\), that is, holey self-orthogonal Latin squares with a symmetric orthogonal mate, consists of three mutually orthogonal Latin squares \(A\), \(B\), and \(C\) of order \(hn\), each having \(n\) holes of size \(h\) that partition the element set, such that \(B=A^T\) and \(C=C^T\). The authors improve the known result for \(h=2\) and show that an \(\text{HSOLSSOM}(2^n)\) exists for \(n\geq 5\) except possibly for \(n\in E=\{8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 28, 32\}\). They then show that an \(\text{HSOLSSOM}(h^n)\) exists for \(n\geq 5\) except possibly for \(n\in E\) if \(h\equiv 2\text{ mod}(4)\) and \(h\geq 10\), and that it exists for \(n\geq 5\) if \(h\equiv 0\text{ mod}(4)\).
scientific article; zbMATH DE number 943215
scientific article