Lengths of divisible codes: the missing cases

Abstract

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.