A Note on the Cross-Correlation of Costas Permutations

Abstract

We build on the work of Drakakis et al. (2011) on the maximal crosscorrelation 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 ≥ 5, the maximal cross-correlation of the family of the ϕ(p − 1) different Welch Costas permutations of {1, . . . , 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 ϕ 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 ≥ 4 the maximal cross-correlation of a subfamily of Golomb Costas permutations of {1, . . . , 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 ϕ(q − 1) whereas there are ϕ(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.