Parity properties of Costas arrays defined via finite fields

TL;DR

This paper investigates the parity distribution of dots in Costas arrays constructed via finite fields, revealing specific numerical relationships and connections to algebraic number theory, especially for Welch-Costas arrays.

Contribution

It provides a detailed enumeration of dot parity types in Golomb-Costas and Welch-Costas arrays, linking their properties to class numbers of quadratic fields.

Findings

Three parity counts in Golomb-Costas arrays are equal and differ by one from the fourth.

In Welch-Costas arrays, parity counts are equal for primes congruent to 1 mod 4.

For primes congruent to 3 mod 4, parity counts relate to class numbers of quadratic fields.

Abstract

A Costas array of order $n$ is an arrangement of dots and blanks into $n$ rows and $n$ columns, with exactly one dot in each row and each column, the arrangement satisfying certain specified conditions. A dot occurring in such an array is even/even 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 dots. Two types of Costas arrays, known as Golomb-Costas and Welch-Costas arrays, can be defined using finite fields. When $q$ is a power of an odd prime, we enumerate the number of even/even odd/odd, even/odd and odd/even dots in a Golomb-Costas array. We show that three of these numbers are equal and they differ by $\pm 1$ from the fourth. For a Welch-Costas array of order $p-1$, where $p$ is an odd prime, the four numbers above are all equal to $(p-1)/4$ when $p\equiv 1\pmod{4}$, but when $p\equiv 3\pmod{4}$, we show that the four numbers are defined in terms of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$, and thus behave in a much less predictable manner.