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!
|
Table of 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.