Lengths of divisible codes: the missing cases

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