On Maximal Partial Costas Latin Squares

Abstract

A Costas array of order $n$ is an $n\times n$ permutation matrix with the property that all of the $n(n-1)/2$ line segments between pairs of $1$’s differ in length or in slope. A Costas latin square of order $n$ is an $n\times n$ latin square where for each symbol $k$, with $1\le k\le n$, the cells containing $k$ determine a Costas array. The existence of a Costas latin square of side $n$ is equivalent to the existence of $n$ mutually disjoint Costas arrays. In 2012, Dinitz, Östergird, and Stinson enumerated all Costas latin squares of side $n\le 27$. In this brief note, a sequel to that paper, we extend this search to sides $n=28$ and $29$. In addition, we determine the sizes of maximal sets of disjoint Costas latin squares of side $n$ for $n\le 29$.