On the Lengths of Divisible Codes

In this article, the effective lengths of all q^{r} -divisible linear codes over \mathbb {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 \mathbb {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 \sigma q^{r} , where \sigma denotes the cross-sum of the S_{q}(r) -adic expansion of n . This result has applications in Galois geometries. A recent theorem of Năstase 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.