A Note on the Cross-Correlation of Costas Permutations

TL;DR

This paper determines the maximum cross-correlation values for Welch and Golomb Costas permutation families, settling some conjectures and providing explicit formulas based on prime divisors.

Contribution

It proves exact maximal cross-correlation bounds for Welch and Golomb Costas permutations, confirming conjectures and refining previous bounds for smaller families.

Findings

Maximal cross-correlation for Welch Costas permutations is (p-1)/t.

Maximal cross-correlation for Golomb Costas permutations is (q-1)/t-1.

Results depend on smallest prime divisors of related numbers.

Abstract

We build on the work of Drakakis et al. (2011) on the maximal cross-correlation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results. First, for a prime $p\ge 5$, the maximal cross-correlation of the family of the $\varphi(p-1)$ different Welch Costas permutations of $\{1,\ldots,p-1\}$ is $(p-1)/t$, where $t$ is the smallest prime divisor of $(p-1)/2$ if $p$ is not a safe prime and at most $1+p^{1/2}$ otherwise. Here $\varphi$ denotes Euler's totient function and a prime $p$ is a safe prime if $(p-1)/2$ is also prime. Second, for a prime power $q\ge 4$ the maximal cross-correlation of a subfamily of Golomb Costas permutations of $\{1,\ldots,q-2\}$ is $(q-1)/t-1$ if $t$ is the smallest prime divisor of $(q-1)/2$ if $q$ is odd and of $q-1$ if $q$ is even provided that $(q-1)/2$ and $q-1$ are not prime, and at most $1+q^{1/2}$ otherwise. Note that we consider a smaller family than Drakakis et al. Our family is of size $\varphi(q-1)$ whereas there are $\varphi(q-1)^2$ different Golomb Costas permutations. The maximal cross-correlation of the larger family given in the tables of Drakakis et al. is larger than our bound (for the smaller family) for some $q$.