2020/06/23 by Gomez-Perez, Domingo, Winterhof, Arne · 1 citation
#05A05 #05B15 #05B20 #94A05 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2006.12820
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+p1/2 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+q1/2 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.