On easily computable indecomposable dimension group algebras, and group codes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909163472289792 |
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| author | García-Claro, E. J. |
| author_facet | García-Claro, E. J. |
| contents | An easily computable dimension (or ECD) group code in the group algebra $\mathbb{F}_{q}G$ is an ideal of dimension less than or equal to $p=char(\mathbb{F}_{q})$ that is generated by an idempotent. This paper introduces an easily computable indecomposable dimension (or ECID) group algebra as a finite group algebra for which all group codes generated by primitive idempotents are ECD. Several characterizations are given for these algebras. In addition, some arithmetic conditions to determine whether a group algebra is ECID are presented, in the case it is semisimple. In the non-semisimple case, these algebras have finite representation type where the Sylow $p$-subgroups of the underlying group are simple. The dimension and some lower bounds for the minimum Hamming distance of group codes in these algebras are given together with some arithmetical tests of primitivity of idempotents. Examples illustrating the main results are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05775 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On easily computable indecomposable dimension group algebras, and group codes García-Claro, E. J. Representation Theory Information Theory 20C05, 11T71, 94B60 An easily computable dimension (or ECD) group code in the group algebra $\mathbb{F}_{q}G$ is an ideal of dimension less than or equal to $p=char(\mathbb{F}_{q})$ that is generated by an idempotent. This paper introduces an easily computable indecomposable dimension (or ECID) group algebra as a finite group algebra for which all group codes generated by primitive idempotents are ECD. Several characterizations are given for these algebras. In addition, some arithmetic conditions to determine whether a group algebra is ECID are presented, in the case it is semisimple. In the non-semisimple case, these algebras have finite representation type where the Sylow $p$-subgroups of the underlying group are simple. The dimension and some lower bounds for the minimum Hamming distance of group codes in these algebras are given together with some arithmetical tests of primitivity of idempotents. Examples illustrating the main results are presented. |
| title | On easily computable indecomposable dimension group algebras, and group codes |
| topic | Representation Theory Information Theory 20C05, 11T71, 94B60 |
| url | https://arxiv.org/abs/2404.05775 |