Self-dual codes with group actions and invariants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910188732153856 |
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| author | Takabayashi, Futo |
| author_facet | Takabayashi, Futo |
| contents | In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden's theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the concept of $G$-dual codes for codes invariant under a permutation group $G$, referred to as $G$-codes. We then present several generalizations of Atsumi's MacWilliams identity (Atsumi, 1995; Chakraborty and Miezaki, 2023) for $G$-codes over finite rings with respect to general bilinear forms. Furthermore, we establish a $G$-analogue of the MacWilliams identity for $G$-full weight enumerators and introduce the notions of $G$-quadratic maps and $G$-representations for twisted modules, twisted rings, quadratic pairs, and form rings. By defining transformation groups for $G$-full weight enumerators, we extend the theory of Clifford--Weil groups (Nebe, Rains, and Sloane, 2004, 2006). Finally, we provide generalizations of Gleason-type theorems for these weight enumerators, demonstrating that the $G$-full weight enumerators of $G$-self-dual and $G$-isotropic codes are invariant under the Clifford--Weil groups and span the invariant subspaces of these groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_02533 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Self-dual codes with group actions and invariants Takabayashi, Futo Combinatorics Group Theory Rings and Algebras 94B05 (Primary) 94B60 (Secondary) In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden's theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the concept of $G$-dual codes for codes invariant under a permutation group $G$, referred to as $G$-codes. We then present several generalizations of Atsumi's MacWilliams identity (Atsumi, 1995; Chakraborty and Miezaki, 2023) for $G$-codes over finite rings with respect to general bilinear forms. Furthermore, we establish a $G$-analogue of the MacWilliams identity for $G$-full weight enumerators and introduce the notions of $G$-quadratic maps and $G$-representations for twisted modules, twisted rings, quadratic pairs, and form rings. By defining transformation groups for $G$-full weight enumerators, we extend the theory of Clifford--Weil groups (Nebe, Rains, and Sloane, 2004, 2006). Finally, we provide generalizations of Gleason-type theorems for these weight enumerators, demonstrating that the $G$-full weight enumerators of $G$-self-dual and $G$-isotropic codes are invariant under the Clifford--Weil groups and span the invariant subspaces of these groups. |
| title | Self-dual codes with group actions and invariants |
| topic | Combinatorics Group Theory Rings and Algebras 94B05 (Primary) 94B60 (Secondary) |
| url | https://arxiv.org/abs/2605.02533 |