Gray arrays and Gray tori (Q1185897)

A Gray array is an \(m\times n\) array of elements chosen from a finite alphabet such that adjacent rows differ in exactly one position and adjacent columns differ in exactly one position. The author proves that \(m\times n\) Gray arrays exist for \(m,n\geq 2\) if and only if \(| m- n|\leq 1\) whenever \(\min\{m,n\}\) is even, and if and only if \(| m- n|\leq 2\) otherwise. He constructs \(n\times n\) and \((n+1)\times n\) Gray arrays for every \(n\geq 1\) and \(m\times(m+2)\) Gray arrays for \(m\) odd. A Gray torus is a Gray array such that both the first and last rows are considered adjacent and the first and last columns are considered adjacent. It is shown that a Gray array \(A\) is a Gray torus if and only if \(A\) is a \(2m\times 2m\) square array and equivalent to a canonical array.