A database of constructions of Hadamard matrices
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914014695522304 |
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| author | Cati, Matteo Pasechnik, Dmitrii V. |
| author_facet | Cati, Matteo Pasechnik, Dmitrii V. |
| contents | Hadamard matrices of order $n$ are conjectured to exist whenever $n$ is $1$, $2$, or a multiple of $4$; a similar conjecture exists for skew Hadamard matrices. We provide constructions covering orders $\le 1208$ of all known Hadamard and skew Hadamard matrices in the open-source software SageMath. This allowed us to verify the correctness of results given in the literature. Within this range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix.
We also produce the up to date tables, for $n \le 2999$ (resp. $n\le 999$ for skew case), of the minimum exponents $m$ such that a (skew) Hadamard matrix of order $2^m n$ is known, improving over 100 entries in the previously published sources. We explain how tables' entries are related to Riesel numbers. As a by-product of the latter, we show that the Paley constructions of (skew-)Hadamard matrices do not work for the order $2^m 509203$, for any $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18897 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A database of constructions of Hadamard matrices Cati, Matteo Pasechnik, Dmitrii V. Combinatorics Discrete Mathematics 05B20, 15B34, 05-04 J.2 Hadamard matrices of order $n$ are conjectured to exist whenever $n$ is $1$, $2$, or a multiple of $4$; a similar conjecture exists for skew Hadamard matrices. We provide constructions covering orders $\le 1208$ of all known Hadamard and skew Hadamard matrices in the open-source software SageMath. This allowed us to verify the correctness of results given in the literature. Within this range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix. We also produce the up to date tables, for $n \le 2999$ (resp. $n\le 999$ for skew case), of the minimum exponents $m$ such that a (skew) Hadamard matrix of order $2^m n$ is known, improving over 100 entries in the previously published sources. We explain how tables' entries are related to Riesel numbers. As a by-product of the latter, we show that the Paley constructions of (skew-)Hadamard matrices do not work for the order $2^m 509203$, for any $m$. |
| title | A database of constructions of Hadamard matrices |
| topic | Combinatorics Discrete Mathematics 05B20, 15B34, 05-04 J.2 |
| url | https://arxiv.org/abs/2411.18897 |