On some properties of Costas arrays generated via finite fields

We prove that Welch-constructed Costas arrays are in general not symmetric and that the Golomb-constructed ones are symmetric in two cases only, namely the Lem-pel one and a (rare) second one leading to the construction of dense Golomb rulers. Finally, we look into the (hard) problem of the number of fixed points of a Welch-constructed Costas array and formulate a conjecture.