Some $3$-designs invariant under $2.PΣL(2,49).$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910538981703680 |
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| author | Shi, Minjia Liu, Ruowen Solé, Patrick |
| author_facet | Shi, Minjia Liu, Ruowen Solé, Patrick |
| contents | We construct a ternary [49,25,7] code from the row span of a Jacobsthal matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the sense of van Lint and MacWilliams (1978). These codes are the abelian generalizations of the quadratic residue (QR) codes which are cyclic. The union of the [50,25,8] extension of the said code and its dual supports a 3-(50,14,1248) design. The automorphism group of the latter design is a double cover of the permutation part of the automorphism group of the [50,25,8] code, which is isomorphic to $PΣL(2,49).$ Other weights in this code, other GQR codes, and other QR codes yield other 3-designs by the same process. A simple group action argument is provided to explain this behaviour of isodual codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some $3$-designs invariant under $2.PΣL(2,49).$ Shi, Minjia Liu, Ruowen Solé, Patrick Information Theory Combinatorics 94 B15, 05 B05 We construct a ternary [49,25,7] code from the row span of a Jacobsthal matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the sense of van Lint and MacWilliams (1978). These codes are the abelian generalizations of the quadratic residue (QR) codes which are cyclic. The union of the [50,25,8] extension of the said code and its dual supports a 3-(50,14,1248) design. The automorphism group of the latter design is a double cover of the permutation part of the automorphism group of the [50,25,8] code, which is isomorphic to $PΣL(2,49).$ Other weights in this code, other GQR codes, and other QR codes yield other 3-designs by the same process. A simple group action argument is provided to explain this behaviour of isodual codes. |
| title | Some $3$-designs invariant under $2.PΣL(2,49).$ |
| topic | Information Theory Combinatorics 94 B15, 05 B05 |
| url | https://arxiv.org/abs/2407.16310 |