On the lengths of divisible codes
Michael Kiermaier, Sascha Kurz · EPub Bayreuth (University of Bayreuth) · 2019
In this article, the effective lengths of all q r -divisible linear codes over F 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 F 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 σ q r , where σ denotes the cross-sum of the S q (r)-adic expansion of n.This result has applications in Galois geometries.A recent theorem of Nȃstase 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.