Lengths of divisible codes

Titelangaben

Körner, Theresa ; Kurz, Sascha:
Lengths of divisible codes.
2023
Veranstaltung: ALgebraic and combinatorial methods for COding and CRYPTography , 20.-24.02.2023 , Marseille, Frankreich.
(Veranstaltungsbeitrag: Kongress/Konferenz/Symposium/Tagung , Vortrag )

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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 are divisible by Delta. The study of divisible codes was initiated by Harold Ward. Linear codes meeting the Griesmer bound in
many cases have to admit a relatively large divisibility constant Delta.

The possible effective lengths of q^r-divisible codes
have been completely characterized for each prime power q and each nonnegative integer r. An implication of these results are upper bound for partial spreads.

More and more applications of divisible codes emerged in the last years, e.g. upper bounds for so-called subspace codes. Noting that the known characterization result for the possible (effective) lengths of q^r-divisible codes involves quite large point multiplicities on the constructive side, there is quite some need for more refined results taking other parameters like the maximum possible point multiplicities or the dimension. Also the restriction
that the exponent r in the divisibility constant Delta = q^r has to be an integer is not always met in the applications. In this talk I present some partial results on the possible effective lengths of divisible codes with extra constraints.