Multidimensional Costas Arrays and Their Periodicity

TL;DR

This paper introduces a new higher-dimensional definition for Costas arrays that works in any dimension, presents non-existence results for periodic multidimensional arrays, and conjectures broader applicability.

Contribution

It proposes a novel multidimensional Costas array definition and extends non-existence results to higher dimensions, generalizing previous two-dimensional findings.

Findings

Three-dimensional arrays with periodic Costas property must have minimal order

Non-existence results for certain multidimensional Costas arrays

Conjecture extending two-dimensional results to arbitrary dimensions

Abstract

A novel higher-dimensional definition for Costas arrays is introduced. This definition works for arbitrary dimensions and avoids some limitations of previous definitions. Some non-existence results are presented for multidimensional Costas arrays preserving the Costas condition when the array is extended periodically throughout the whole space. In particular, it is shown that three-dimensional arrays with this property must have the least possible order; extending an analogous two-dimensional result by H. Taylor. Said result is conjectured to extend for Costas arrays of arbitrary dimensions.