A duality for nonabelian group codes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909121649836032 |
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| author | Wentworth-Nice, Prairie |
| author_facet | Wentworth-Nice, Prairie |
| contents | In 1962, Jesse MacWilliams published a set of formulas for linear and abelian group codes that among other applications, were incredibly valuable in the study of self-dual codes. Now called the MacWilliams Identities, her results relate the weight enumerator and complete weight enumerator of a code to those of its dual code. A similar set of MacWilliams identities has been proven to exist for many other types of codes. In 2013, Dougherty, Solé, and Kim published a list of fundamental open questions in coding theory. Among them, Open Question 4.3: "Is there a duality and MacWilliams formula for codes over non-Abelian groups?" In this paper, we propose a duality for nonabelian group codes in terms of the irreducible representations of the group. We show that there is a Greene's Theorem and MacWilliams Identities which hold for this notion of duality. When the group is abelian, our results are equivalent to existing formulas in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_17597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A duality for nonabelian group codes Wentworth-Nice, Prairie Information Theory Group Theory 94B60 In 1962, Jesse MacWilliams published a set of formulas for linear and abelian group codes that among other applications, were incredibly valuable in the study of self-dual codes. Now called the MacWilliams Identities, her results relate the weight enumerator and complete weight enumerator of a code to those of its dual code. A similar set of MacWilliams identities has been proven to exist for many other types of codes. In 2013, Dougherty, Solé, and Kim published a list of fundamental open questions in coding theory. Among them, Open Question 4.3: "Is there a duality and MacWilliams formula for codes over non-Abelian groups?" In this paper, we propose a duality for nonabelian group codes in terms of the irreducible representations of the group. We show that there is a Greene's Theorem and MacWilliams Identities which hold for this notion of duality. When the group is abelian, our results are equivalent to existing formulas in the literature. |
| title | A duality for nonabelian group codes |
| topic | Information Theory Group Theory 94B60 |
| url | https://arxiv.org/abs/2402.17597 |