A linear code C over \({\mathbb {F}}_q\) is called \(\Delta \)-divisible if the Hamming weights \({\text {wt}}(c)\) 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 in Kiermaier and Kurz (IEEE Trans Inf Theory 66(7):4051–4060, 2020). The study of \(\Delta \)-divisible codes was initiated by Harold Ward (Archiv der Mathematik 36(1):485–494, 1981). If t divides \(\Delta \) but is coprime to q, then each \(\Delta \)-divisible code C over \({\mathbb {F}}_q\) is the t-fold repetition of a \(\Delta /t\)-divisible code. Here we determine the possible effective lengths of \(p^r\)-divisible codes over finite fields of characteristic p, where \(r\in {\mathbb {N}}\) but \(p^r\) is not a power of the field size, i.e., the missing cases.

如果所有码字的汉明权重\ ({\text {wt}}(c)\) \({\mathbb {F}}_q\)上的线性码C称为\(\Delta \)可整除c \in C\)可以被\(\Delta \)整除。Kiermaier 和 Kurz 中的每个素数幂q和每个非负整数r的\(q^r\)可整除码的可能有效长度已被完全表征（IEEE Trans Inf Theory 66(7):4051–4060, 2020 ）。\(\Delta \)可整除码的研究由 Harold Ward 发起（Archiv der Mathematik 36(1):485–494, 1981）。如果t整除\(\Delta \)但与q互质，则\ ({\mathbb {F}}_q\)上的每个\(\Delta \)整除代码C是\ ( \Delta /t\) -可整除的代码。这里我们确定特征p的有限域上的\(p^r\)可整除码的可能有效长度，其中\(r\in {\mathbb {N}}\)但\(p^r\)不是字段大小的幂，即缺失的案例。