A Note on the Cross-Correlation of Costas Permutations
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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≥ 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.
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