Five mutually orthogonal latin squares of orders 24 and 40
An \((r,n)\)-difference matrix \(D\), over an abelian group \(G\) of order \(n\), is an \(r\times n\) matrix such that the difference of any two distinct rows results in a vector whose set of components equals \(G\). Let \(N(n)\) denote the maximum number of mutually orthogonal latin squares of order \(n\). The existence of an \((r,n)\)-difference matrix implies that \(N(n)\geq r-1\). The author uses a computer to construct \((6,24)\)-difference matrix over \(Z_6 \oplus Z_2 \oplus Z_2\) and \((6,40)\)-difference matrices over \(Z_{10} \oplus Z_2 \oplus Z_2\), thereby establishing that \(N(24)\geq 5\) and \(N(40)\geq 5\).
scientific article; zbMATH DE number 769322