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arXiv · 1703.08291

Projective divisible binary codes

Daniel Heinlein·Thomas Honold·Michael Kiermaier·Sascha Kurz·Alfred Wassermann
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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 $\operatorname{PG}(v-1,\mathbb{F}_q)$ corresponds to a $q^r$-divisible code with $k\leq v$. In this paper we provide an introduction to this problem and report on new results for $q=2$.

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Daniel Heinlein, Thomas Honold, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. 2017-03-24. Projective divisible binary codes. https://arxiv.org/abs/1703.08291

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