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 σq
^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 Nästase 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.