Packing Costas Arrays

TL;DR

This paper investigates Costas Latin squares, providing a classification up to order 27, and explores related structures, confirming the conjecture that no such squares exist for odd orders n ≥ 3.

Contribution

It offers a complete classification of Costas Latin squares up to order 27 and verifies the non-existence for odd orders n ≥ 3, introducing related combinatorial structures.

Findings

No Costas Latin squares for odd n ≥ 3 up to order 27

Complete classification of Costas Latin squares up to order 27

Analysis of related structures like near Costas Latin squares

Abstract

A 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.