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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2411.07785 |
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| _version_ | 1866912116085096448 |
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| author | Gow, Rod McGuire, Gary |
| author_facet | Gow, Rod McGuire, Gary |
| contents | Let $F$ be any field containing the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that all powers of $x$ that appear in $L$ with nonzero coefficient have exponent a power of $q$. It is well known that given any ordinary polynomial $f$ in $F[x]$, there exists a $q$-polynomial that is divisible by $f$. We study the smallest degree of such a $q$-polynomial. This is equivalent to studying the $\mathbb{F}_q$-span of the roots of $f$ in a splitting field. We relate this quantity to the representation theory of the Galois group of $f$. As an application we give a simultaneous construction of the binary Golay code of length 24, and the Steiner system on 24 points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07785 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linearization of polynomials in prime characteristic, with applications to the Golay code and Steiner system Gow, Rod McGuire, Gary Number Theory Let $F$ be any field containing the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that all powers of $x$ that appear in $L$ with nonzero coefficient have exponent a power of $q$. It is well known that given any ordinary polynomial $f$ in $F[x]$, there exists a $q$-polynomial that is divisible by $f$. We study the smallest degree of such a $q$-polynomial. This is equivalent to studying the $\mathbb{F}_q$-span of the roots of $f$ in a splitting field. We relate this quantity to the representation theory of the Galois group of $f$. As an application we give a simultaneous construction of the binary Golay code of length 24, and the Steiner system on 24 points. |
| title | Linearization of polynomials in prime characteristic, with applications to the Golay code and Steiner system |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.07785 |