# Costas cubes

## TL;DR

This paper introduces three-dimensional Costas cubes, explores their existence for small orders, and presents constructions for infinite families, expanding the combinatorial understanding of Costas arrays into higher dimensions.

## Contribution

It defines Costas cubes as 3D analogs of Costas arrays, analyzes their existence for small orders, and provides new infinite construction methods.

## Key findings

- Costas cubes exist for all orders up to 29 except 18 and 19.
- A significant number of Costas arrays are projections of Costas cubes.
- Four infinite families of Costas cubes are constructed.

## 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.