Costas cubes

Abstract

A Costas array is a permutation array for which the vectors joining pairs of $1$s are all distinct. We propose a new three-dimensional combinatorial object related to Costas arrays: an order $n$ Costas cube is an array $(d_{i,j,k})$ of size $n\times n\times n$ over ${\mathbb{Z}}_2$ for which each of the three projections of the array onto two dimensions, namely $({\sum }_i{d}_{i,j,k})$ and $({\sum }_j{d}_{i,j,k})$ and $({\sum }_k{d}_{i,j,k})$, is an order $n$ Costas array. We determine all Costas cubes of order at most $29$, showing that Costas cubes exist for all these orders except $18$ and $19$ and that a significant proportion of the Costas arrays of certain orders occur as projections of Costas cubes. We then present constructions for four infinite families of Costas cubes.

12 pages, 1 figure. Theorem 11 introduces two further infinite families of Costas cubes