Lengths of divisible codes -- the missing cases

Abstract

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.