On the Hops Present in Costas Permutations

Given that frequency-hopping filters cannot easily implement big frequency hops instantaneously, those Costas permutations are determined in which the maximal frequency hop prescribed is as small as possible, as well as those that do contain the maximal hop possible, and are, consequently, less suitable for applications. It turns out that exponential Welch permutations not only lead in general to the smallest hops, but are also relatively easy to study, as a closed formula exists for the maximal hop. Through extensive collection of data for logarithmic Welch and Golomb permutations, on the other hand, it is found that: a) these two families behave (almost) identically; and that b) their maximal hops do not get as small as in exponential Welch permutations.