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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.01502 |
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| _version_ | 1866909447545159680 |
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| author | Willenborg, Nadja |
| author_facet | Willenborg, Nadja |
| contents | Codes in the generalized quaternion group algebra $\mathbb{F}_q[Q_{4n}]$ are considered. Restricting to char$\mathbb{F}_q \nmid 4n$ the structure of an arbitrary code $C \subseteq \mathbb{F}_q[Q_{4n}]$ is described via the Wedderburn decomposition. Moreover it is known that in this case every code $C \subseteq \mathbb{F}_q[Q_{4n}]$ has a generating idempotent $λ\in \mathbb{F}_q[Q_{4n}]$. Given the generating idempotent of a code $C$ we determine the different components in its decomposition $C \cong \bigoplus_{j=1}^{r+s}C_j \oplus \bigoplus_{i=1}^{k+t}C'_{i}.$ Afterwards we apply this result to describe the blocks of codes induced by cyclic group codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Block components of generalized quaternion group codes Willenborg, Nadja Information Theory Combinatorics Codes in the generalized quaternion group algebra $\mathbb{F}_q[Q_{4n}]$ are considered. Restricting to char$\mathbb{F}_q \nmid 4n$ the structure of an arbitrary code $C \subseteq \mathbb{F}_q[Q_{4n}]$ is described via the Wedderburn decomposition. Moreover it is known that in this case every code $C \subseteq \mathbb{F}_q[Q_{4n}]$ has a generating idempotent $λ\in \mathbb{F}_q[Q_{4n}]$. Given the generating idempotent of a code $C$ we determine the different components in its decomposition $C \cong \bigoplus_{j=1}^{r+s}C_j \oplus \bigoplus_{i=1}^{k+t}C'_{i}.$ Afterwards we apply this result to describe the blocks of codes induced by cyclic group codes. |
| title | Block components of generalized quaternion group codes |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2501.01502 |