On the degrees of freedom of Costas permutations and other constraints

The number of degrees of freedom of Costas permutations is considered, and found to be surprisingly small, while partial results about the degrees of freedom of Golomb and Welch Costas permutations are proved. For Golomb Costas permutations, in particular, the curious observation is made that arbitrarily long sequences of distinct positive integers seem to exist, with the property that two or more Golomb Costas permutations, constructed in a suitably large field, start with such a sequence; other types of constraints, related to their cycle structure, are studied; and finally it is shown that, in any extension field containing non-quadratic subfields, Lempel Costas permutations are obtainable through the iterated composition of other Golomb Costas permutations.