Circular Costas maps: a multidimensional analog of circular Costas sequences

TL;DR

This paper develops a comprehensive theoretical framework for multidimensional Costas structures, connecting various sequence types and proving several longstanding conjectures in the field.

Contribution

It introduces a unifying multidimensional framework and proves multiple conjectures related to Costas arrays, sequences, and polynomials, extending classical results to higher dimensions.

Findings

Proved several conjectures on multidimensional periodic Costas arrays.

Established a multidimensional extension of the shifting property for Costas sequences.

Linked Costas polynomials over extension fields to multidimensional Costas structures.

Abstract

A unifying theoretical framework is presented, in which the connections among Costas sequences, circular Costas sequences, Costas polynomials, the shifting property, and Welch sequences are extended to the multidimensional context. Several conjectures on multidimensional periodic Costas arrays by J. Ortiz-Ubarri et al. are proved. Furthermore, a conjecture on Costas polynomials over extension fields presented by Muratovic-Ribic et al. is showed to be a multidimensional extension of a conjecture by Golomb and Moreno on circular Costas sequences. A weaker version of said conjecture is proved by considering a multidimensional extension of the shifting Costas property defined by O. Moreno.