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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 $kleq v$. In this paper we provide an introduction to this problem and report on new results for $q=2$.
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A projective linear code over $mathbb{F}_q$ is called $Delta$-divisible if all weights of its codewords are divisible by $Delta$. Especially, $q^r$-divisible projective linear codes, where $r$ is some integer, arise in many applications of collections of subspaces in $mathbb{F}_q^v$. One example are upper bounds on the cardinality of partial spreads. Here we survey the known results on the possible lengths of projective $q^r$-divisible linear codes.
We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over $mathbb{F}_q$ and their classification. Through a mix of linear programming, combinatorial techniques and computer enumeration, we investigate the structural properties of these codes. In this way, we can prove a reduction theorem for a generalization of the cylinder conjecture, show some instances where it does not hold and prove its validity for small values of $q$. In particular, we correct a flawed proof for the original cylinder conjecture for $q = 5$ and present the first proof for $q = 7$.
We classify all $q$-ary $Delta$-divisible linear codes which are spanned by codewords of weight $Delta$. The basic building blocks are the simplex codes, and for $q=2$ additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight $4$ have been classified, which is the case $q=2$ and $Delta=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $Delta$-divisible codes of length $4Delta$ in the projective case.
We classify 8-divisible binary linear codes with minimum distance 24 and small length. As an application we consider the codes associated to nodal sextics with 65 ordinary double points.
Given any linear code $C$ over a finite field $GF(q)$ we show how $C$ can be described in a transparent and geometrical way by using the associated Bruen-Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes.
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