On the Lengths of Divisible Codes

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 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 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.