On the Cross-Correlation of Golomb Costas Permutations

• Published in: IEEE transactions on information theory Vol. 70; no. 11; pp. 7848 - 7852
• Main Authors: Liu, Huaning, Winterhof, Arne
• Format: Journal Article
• Language: English
• Published: IEEE 01.11.2024
• Subjects: Costas arrays; cross-correlation; finite fields; Golomb’s construction; Mathematics; Physics; Polynomials; Postal services; Radar applications; Radon; Sonar applications; Szemerédi-Trotter theorem; Weil bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2024.3460189

In the most interesting case of safe prime powers q, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of <inline-formula> <tex-math notation="LaTeX">\{1,2,\ldots,q-2\} </tex-math></inline-formula> of size <inline-formula> <tex-math notation="LaTeX">\varphi (q-1) </tex-math></inline-formula> has maximal cross-correlation of order of magnitude at most <inline-formula> <tex-math notation="LaTeX">q^{1/2} </tex-math></inline-formula>. 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 Szemerédi-Trotter theorem for finite fields.