Parity properties of Costas arrays defined via finite fields
| dc.creator | Drakakis, Konstantinos | |
| dc.creator | Gow, Rod | |
| dc.creator | rickard, Scott | |
| dc.date | 2007-06-22 | |
| dc.date.accessioned | 2026-07-07T08:11:53Z | |
| dc.date.available | 2026-07-07T08:11:53Z | |
| dc.description | 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. | |
| dc.description | To appear in Advances in Mathematics of Communications | |
| dc.identifier | https://arxiv.org/abs/0706.3313 | |
| dc.identifier | http://arxiv.org/abs/0706.3313 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132268 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.title | Parity properties of Costas arrays defined via finite fields | |
| dc.type | text |