On the lengths of divisible codes

Titelangaben

Kiermaier, Michael ; Kurz, Sascha:
On the lengths of divisible codes.
In: IEEE Transactions on Information Theory. Bd. 66 (2020) Heft 7 . - S. 4051-4060.
ISSN 0018-9448
DOI: https://doi.org/10.1109/TIT.2020.2968832

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Abstract

In this article, the effective lengths of all q^r-divisible linear codes over GF(q) with a non-negative integer r are determined. For that purpose, the S_q(r)-adic expansion of an integer n is introduced. It is shown that there exists a q^r-divisible GF(q)-linear code of effective length n if and only if the leading coefficient of the S_q(r)-adic expansion of n is non-negative. Furthermore, the maximum weight of a q^r-divisible code of effective length n is at most the cross-sum of the S_q(r)-adic expansion of n.

This result has applications in Galois geometries.
A recent theorem of Nastase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.