On the lengths of divisible codes

Abstract

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 N{ă}stase 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.

17 pages, typos corrected; the paper was originally named "An improvement of the Johnson bound for subspace codes"

References in corpus (4)

• Improved upper bounds for partial spreads
• Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound
• Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6
• The lengths of projective triply-even binary codes

Cited by in corpus (5)

• Bounds for flag codes
• Affine vector space partitions
• No projective 16-divisible binary linear code of length 131 exists
• Multilevel inserting constructions for constant dimension subspace codes
• Lengths of divisible codes -- the missing cases