# American Institute of Mathematical Sciences

August  2007, 1(3): 321-330. doi: 10.3934/amc.2007.1.321

## Parity properties of Costas arrays defined via finite fields

 1 School of Electrical, Electronic & Mechanical Engineering, University College Dublin, Belﬁeld, Dublin 4, Ireland, Ireland 2 School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

Received  December 2006 Revised  June 2007 Published  July 2007

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 $p\equiv 1(\mod 4)$, but when $p\equiv 3(\mod 4)$, 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.
Citation: Konstantinos Drakakis, Rod Gow, Scott Rickard. Parity properties of Costas arrays defined via finite fields. Advances in Mathematics of Communications, 2007, 1 (3) : 321-330. doi: 10.3934/amc.2007.1.321
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