A linear code $C$ over $\mathbb{F}_q$ is called $\Delta$-divisible if the Hamming weights $\operatorname{wt}(c)$ 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.

翻译：设 $\mathbb{F}_q$ 上的线性码 $C$ 若满足所有码字 $c \in C$ 的汉明重量 $\operatorname{wt}(c)$ 均被 $\Delta$ 整除，则称为 $\Delta$-可整除码。对于每个素数幂 $q$ 和每个非负整数 $r$，$q^r$-可整除码的可能有效长度已被完全刻画。$\Delta$ 可整除码的研究由 Harold Ward 开创。若 $c$ 整除 $\Delta$ 但与 $q$ 互素，则 $\mathbb{F}_q$ 上的每个 $\Delta$-可整除码 $C$ 都是 $\Delta/c$-可整除码的 $c$ 重重复。本文确定了特征为 $p$ 的有限域上 $p^r$-可整除码的可能有效长度，其中 $p\in\mathbb{N}$ 但 $p^r$ 不是域大小的幂，即缺失情况。