Lengths of divisible codes - the missing cases

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Kurz, Sascha:
Lengths of divisible codes - the missing cases.
Bayreuth , 2023 . - 12 S.

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Abstract

A linear code C over GF(q) is called Δ-divisible if the Hamming weights wt(c) of all codewords c in C are divisible by Δ. 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 Δ divisible codes was initiated by Harold Ward. If c divides Δ but is coprime to q, then each Δ-divisible code C over GF(q) is the c-fold repetition of a Δ/c-divisible code. Here we determine the possible effective lengths of p^r-divisible codes over finite fields of characteristic p, where r is an integer but p^r is not a power of the field size, i.e., the missing cases.