Lengths of divisible codes : the missing cases

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Kurz, Sascha:
Lengths of divisible codes : the missing cases.
In: Designs, Codes and Cryptography. Bd. 92 (2024) . - S. 2367-2378.
ISSN 1573-7586
DOI: https://doi.org/10.1007/s10623-024-01398-7

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Abstract

A linear code C over GF(q) is called Delta-divisible if the Hamming weights 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 Δ divisible codes was initiated by Harold Ward. If c divides Delta but is coprime to q, then each Delta-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.