On catastrophicity of convolutional codes and their encoders over $\Z_{p^r}$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918137510756352 |
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| author | Oued, Mohammed El |
| author_facet | Oued, Mohammed El |
| contents | This paper investigates the existence of minimal $p$-encoders for convolutional codes over $\mathbb{Z}_{p^r}$, where $p$ is a prime. This addresses a conjecture from \cite{k}, which posits that every such code admits a minimal $p$-encoder, implying that all convolutional codes over $\mathbb{Z}_{p^r}$ are noncatastrophic when input sequences are restricted to coefficients in $\{0, \dots, p-1\}$. Our contributions include the introduction of a new polynomial invariant that characterizes free codes, which enables us to establish a necessary and sufficient condition for a free code over $\mathbb{Z}_{p^r}$ to be noncatastrophic in the usual sense (where input coefficients are from $\mathbb{Z}_{p^r}$). Based on these findings, we affirm the conjecture by providing a constructive method for obtaining a minimal $p$-encoder for any convolutional code over $\mathbb{Z}_{p^r}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On catastrophicity of convolutional codes and their encoders over $\Z_{p^r}$ Oued, Mohammed El Information Theory This paper investigates the existence of minimal $p$-encoders for convolutional codes over $\mathbb{Z}_{p^r}$, where $p$ is a prime. This addresses a conjecture from \cite{k}, which posits that every such code admits a minimal $p$-encoder, implying that all convolutional codes over $\mathbb{Z}_{p^r}$ are noncatastrophic when input sequences are restricted to coefficients in $\{0, \dots, p-1\}$. Our contributions include the introduction of a new polynomial invariant that characterizes free codes, which enables us to establish a necessary and sufficient condition for a free code over $\mathbb{Z}_{p^r}$ to be noncatastrophic in the usual sense (where input coefficients are from $\mathbb{Z}_{p^r}$). Based on these findings, we affirm the conjecture by providing a constructive method for obtaining a minimal $p$-encoder for any convolutional code over $\mathbb{Z}_{p^r}$. |
| title | On catastrophicity of convolutional codes and their encoders over $\Z_{p^r}$ |
| topic | Information Theory |
| url | https://arxiv.org/abs/2509.06670 |