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Register a new userA generalization of the cylinder conjecture for divisible codes
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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 $q = 7$.
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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 $q=2$.
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 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.
In this paper we investigate combinatorial constructions for $w$-cyclic holely group divisible packings with block size three (briefly by $3$-HGDPs). For any positive integers $u,v,w$ with $uequiv0,1~(bmod~3)$, the exact number of base blocks of a maximum $w$-cyclic $3$-HGDP of type $(u,w^v)$ is determined. This result is used to determine the exact number of codewords in a maximum three-dimensional $(utimes vtimes w,3,1)$ optical orthogonal code with at most one optical pulse per spatial plane and per wavelength plane.
A well known class of objects in combinatorial design theory are {group divisible designs}. Here, we introduce the $q$-analogs of group divisible designs. It turns out that there are interesting connections to scattered subspaces, $q$-Steiner systems, design packings and $q^r$-divisible projective sets. We give necessary conditions for the existence of $q$-analogs of group divsible designs, construct an infinite series of examples, and provide further existence results with the help of a computer search. One example is a $(6,3,2,2)_2$ group divisible design over $operatorname{GF}(2)$ which is a design packing consisting of $180$ blocks that such every $2$-dimensional subspace in $operatorname{GF}(2)^6$ is covered at most twice.
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