Parity properties of Costas arrays defined via finite fields

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

Parity properties of Costas arrays defined via finite fields

Konstantinos Drakakis ^1, , Rod Gow ^2, and Scott Rickard ^1,

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

Abstract
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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.

Keywords: Welch construction, class number, permutation arrays., Costas arrays, Golomb construction, parity populations, finite fields.

Mathematics Subject Classification: 11T06, 11R29, 05B2.

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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