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A Note on the Cross-Correlation of Costas Permutations
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 , the maximal cross-correlation of the family of
the different Welch Costas permutations of is
, where is the smallest prime divisor of if is not a
safe prime and at most otherwise. Here denotes Euler's
totient function and a prime is a safe prime if is also prime.
Second, for a prime power the maximal cross-correlation of a
subfamily of Golomb Costas permutations of is if
is the smallest prime divisor of if is odd and of if
is even provided that and are not prime, and at most
otherwise. Note that we consider a smaller family than Drakakis et
al. Our family is of size whereas there are
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