On the parity populations of Welch-constructed Costas arrays

We prove that, in the case of Welch-constructed Costas arrays, the number of dots whose coordinates are both even (ee), both odd (oo), even and odd (eo), and odd and even (oe) are all equal if the prime p used has the property that p mod 4=1; and that, if p mod 4=3, the relation between these 4 quantities can again be determined, and that it involves an unexpected appearance of class numbers.