Packing Costas arrays (Q2895355)

A Costas array of order \(n\) (or side \(n\)) is an \(n \times n\) array of dots and empty cells such that: 1.) there are \(n\) dots and \(n(n-1)\) empty cells, with exactly one dot in each row and column, and 2.) all the segments between pairs of dots differ in length or in slope.NEWLINENEWLINENEWLINETwo Costas arrays of order \(n\) are disjoint if there is no cell in which both arrays have a dot.NEWLINENEWLINENEWLINEA Costas Latin square of order \(n\) is a set of \(n\) disjoint Costas arrays of the same order. Costas Latin squares are studied here from a construction as well as a classification point of view. A complete classification is carried out up to order 27. In this range, we verify the conjecture that there is no Costas Latin square for any odd order \(n > 3\). Various other related combinatorial structures are also considered, including near Costas Latin squares (which are certain packings of near Costas arrays) and Vatican Costas squares.