Lengths of divisible codes -- the missing cases

Abstract

A linear code $C$ over ${\mathbb{F}}_q$ is called $\Delta$-divisible if the Hamming weights $\mathrm{wt}(c)$ of all codewords $c\in C$ are divisible by $\Delta$. 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 $\Delta$ divisible codes was initiated by Harold Ward. If $c$ divides $\Delta$ but is coprime to $q$, then each $\Delta$-divisible code $C$ over $\F_q$ is the $c$-fold repetition of a $\Delta /c$-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.

11 pages, 1 table

References in corpus (1)

• A note on small weight codewords of projective geometric codes and on the smallest sets of even type