A Regularity Property of Golomb-Costas Arrays

A Golomb-Costas array is an arrangement of dots and blanks, defined for each positive integer power of a prime and satisfying certain unusual conditions. A dot occurring in such an array is an even/even position if it occurs in the i-th row and j-th column, where i and j are both even integers, and there are similar definitions of odd/odd, even/odd and odd/even positions for dots. When q is a power of an odd prime, we enumerate the number of even/even, odd/odd, even/odd and odd/even positions for dots in a Golomb-Costas array of order q-2. We show that three of these numbers are equal and they differ by differ by plusmn1 from the fourth. More general Costas arrays do not exhibit this regularity. We also show that if q=r
^t
, where r is a power of a prime and t is an integer greater than 1, any Golomb-Costas array of order q-2 contains in a natural way a Golomb-Costas array of order r-2 which can easily be identified.