Projective divisible binary codes

Abstract

For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over ${\mathbb{F}}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The motivation for studying this problem comes from the theory of partial spreads, or subspace codes with the highest possible minimum distance, since the set of holes of a partial spread of $r$-flats in $\mathrm{PG}(v-1,{\mathbb{F}}_q)$ corresponds to a $q^r$-divisible code with $k\le v$. In this paper we provide an introduction to this problem and report on new results for $q=2$.

10 pages, 3 tables

References in corpus (4)

• A Rank-Metric Approach to Error Control in Random Network Coding
• Partial spreads and vector space partitions
• On triply even binary codes
• Classification of large partial plane spreads in $PG(6,2)$ and related combinatorial objects

Cited by in corpus (10)

• Tables of subspace codes
• Computer classification of linear codes
• On strongly walk regular graphs, triple sum sets and their codes
• LinCode -- computer classification of linear codes
• New LMRD bounds for constant dimension codes and improved constructions
• Classification of linear codes using canonical augmentation
• Heden's bound on the tail of a vector space partition
• Generalized vector space partitions
• On projective $q^r$-divisible codes
• Classification of $\Delta$-divisible linear codes spanned by codewords of weight $\Delta$