## Title data

Kiermaier, Michael ; Kurz, Sascha:

**On the lengths of divisible codes.**

Bayreuth
,
2019
. - 17 p.

## Project information

Project title: |
Project's official title Project's id Integer Linear Programming Models for Subspace Codes and Finite Geometry No information |
---|---|

Project financing: |
Deutsche Forschungsgemeinschaft |

## Abstract in another language

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 sizes of partial spreads follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.