On the cross-correlation of Golomb Costas permutations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913480886452224 |
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| author | Liu, Huaning Winterhof, Arne |
| author_facet | Liu, Huaning Winterhof, Arne |
| contents | In the most interesting case of safe prime powers $q$, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of $\{1,2,\ldots,q-2\}$ of size $φ(q-1)$ has maximal cross-correlation of order of magnitude at most $q^{1/2}$. In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemerédi-Trotter theorem for finite fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_14330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the cross-correlation of Golomb Costas permutations Liu, Huaning Winterhof, Arne Combinatorics Number Theory In the most interesting case of safe prime powers $q$, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of $\{1,2,\ldots,q-2\}$ of size $φ(q-1)$ has maximal cross-correlation of order of magnitude at most $q^{1/2}$. In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemerédi-Trotter theorem for finite fields. |
| title | On the cross-correlation of Golomb Costas permutations |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2408.14330 |