13 research outputs found
Classifying optimal binary subspace codes of length 8, constant dimension 4 and minimum distance 6
The maximum size of a binary subspace code of packet length
, minimum subspace distance , and constant dimension is ,
where the isomorphism types are extended lifted maximum rank distance
codes. In finite geometry terms the maximum number of solids in
, mutually intersecting in at most a point, is .
The result was obtained by combining the classification of substructures with
integer linear programming techniques. This implies that the maximum size
of a binary mixed-dimension code of packet length and minimum
subspace distance is as well.Comment: 17 pages, 3 table
Subspaces intersecting in at most a point
We improve on the lower bound of the maximum number of planes in
\operatorname{PG}(8,q)\cong\F_q^{9} pairwise intersecting in at most a point.
In terms of constant dimension codes this leads to . This result is obtained via a more general
construction strategy, which also yields other improvements.Comment: 4 page
On the lengths of divisible codes
In this article, the effective lengths of all -divisible linear codes
over with a non-negative integer are determined. For that
purpose, the -adic expansion of an integer is introduced. It is
shown that there exists a -divisible -linear code of
effective length if and only if the leading coefficient of the
-adic expansion of is non-negative. Furthermore, the maximum weight
of a -divisible code of effective length is at most ,
where denotes the cross-sum of the -adic expansion of .
This result has applications in Galois geometries. A recent theorem of
N{\u{a}}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.Comment: 17 pages, typos corrected; the paper was originally named "An
improvement of the Johnson bound for subspace codes
q-analogs of group divisible designs
A well known class of objects in combinatorial design theory are {group
divisible designs}. Here, we introduce the -analogs of group divisible
designs. It turns out that there are interesting connections to scattered
subspaces, -Steiner systems, design packings and -divisible projective
sets.
We give necessary conditions for the existence of -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 group divisible design over
which is a design packing consisting of blocks
that such every -dimensional subspace in is covered
at most twice.Comment: 18 pages, 3 tables, typos correcte
Combining subspace codes
In the context of constant--dimension subspace codes, an important problem is
to determine the largest possible size of codes whose codewords
are -subspaces of with minimum subspace distance . Here
in order to obtain improved constructions, we investigate several approaches to
combine subspace codes. This allow us to present improvements on the lower
bounds for constant--dimension subspace codes for many parameters, including
, , and .Comment: 17 pages; construction for A_(10,4;5) was flawe