Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.01947 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916618494279680 |
|---|---|
| author | Kurz, Sascha |
| author_facet | Kurz, Sascha |
| contents | A linear code $C$ over $\mathbb{F}_q$ is called $Δ$-divisible if the Hamming weights $\operatorname{wt}(c)$ of all codewords $c \in C$ are divisible by $Δ$. The possible effective lengths of $q^r$-divisible codes have been completely characterized for each prime power $q$ and each non-negative integer $r$. The study of $Δ$ divisible codes was initiated by Harold Ward. If $c$ divides $Δ$ but is coprime to $q$, then each $Δ$-divisible code $C$ over $\F_q$ is the $c$-fold repetition of a $Δ/c$-divisible code. Here we determine the possible effective lengths of $p^r$-divisible codes over finite fields of characteristic $p$, where $p\in\mathbb{N}$ but $p^r$ is not a power of the field size, i.e., the missing cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01947 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lengths of divisible codes -- the missing cases Kurz, Sascha Combinatorics Information Theory 51E23, 05B40) A linear code $C$ over $\mathbb{F}_q$ is called $Δ$-divisible if the Hamming weights $\operatorname{wt}(c)$ of all codewords $c \in C$ are divisible by $Δ$. The possible effective lengths of $q^r$-divisible codes have been completely characterized for each prime power $q$ and each non-negative integer $r$. The study of $Δ$ divisible codes was initiated by Harold Ward. If $c$ divides $Δ$ but is coprime to $q$, then each $Δ$-divisible code $C$ over $\F_q$ is the $c$-fold repetition of a $Δ/c$-divisible code. Here we determine the possible effective lengths of $p^r$-divisible codes over finite fields of characteristic $p$, where $p\in\mathbb{N}$ but $p^r$ is not a power of the field size, i.e., the missing cases. |
| title | Lengths of divisible codes -- the missing cases |
| topic | Combinatorics Information Theory 51E23, 05B40) |
| url | https://arxiv.org/abs/2311.01947 |