Lengths of divisible codes -- the missing cases

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