On the cross-correlation of Golomb Costas permutations

TL;DR

This paper investigates the cross-correlation properties of Golomb Costas permutations, extending previous results to larger families and establishing bounds and rarity of high cross-correlation instances using advanced mathematical tools.

Contribution

It extends the analysis of cross-correlation bounds to larger families of Golomb Costas permutations and identifies conditions for low cross-correlation.

Findings

Large cross-correlations are rare among Golomb Costas permutations.

A weaker bound on maximal cross-correlation for a larger family is established.

Conditions for small cross-correlation are identified.

Abstract

In the most interesting case of safe prime powers $q$, G\'omez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of $\{1,2,\ldots,q-2\}$ of size $\varphi(q-1)$ has maximal cross-correlation of order of magnitude at most $q^{1/2}$. In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemer\'edi-Trotter theorem for finite fields.