90,681 research outputs found
Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces
In this paper we investigate partial spreads of through the
related notion of partial spread sets of hermitian matrices, and the more
general notion of constant rank-distance sets. We prove a tight upper bound on
the maximum size of a linear constant rank-distance set of hermitian matrices
over finite fields, and as a consequence prove the maximality of extensions of
symplectic semifield spreads as partial spreads of . We prove
upper bounds for constant rank-distance sets for even rank, construct large
examples of these, and construct maximal partial spreads of for a
range of sizes
Linear codes meeting the Griesmer bound, minihypers and geometric applications
Coding theory and Galois geometries are two research areas which greatly influence each other. In this talk, we focus on the link between linear codes meeting the Griesmer bound and minihypers in finite projective spaces. Minihypers are particular (multiple) blocking sets. We present characterization results on minihypers, leading to equivalent characterization results on linear codes meeting the Griesmer bound. Next to being interesting from a coding-theoretical point of view, minihypers also are interesting for geometrical applications. We present results on maximal partial μ-spreads in PG(N, q), (μ + 1)|(N + 1), on minimal μ-covers in PG(N, q), (μ + 1)|(N + 1), on (N − 1)-covers of Q + (2N + 1, q), on partial ovoids and on partial spreads of finite classical polar spaces, and on partial ovoids of generalized hexagons, following from results on minihypers
Projective divisible binary codes
For which positive integers does there exist a linear code
over with all codeword weights divisible by and such
that the columns of a generating matrix of 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 -flats in
corresponds to a -divisible code with . In this paper we provide
an introduction to this problem and report on new results for .Comment: 10 pages, 3 table
The use of blocking sets in Galois geometries and in related research areas
Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems
Classification of large partial plane spreads in and related combinatorial objects
In this article, the partial plane spreads in of maximum possible
size and of size are classified. Based on this result, we obtain the
classification of the following closely related combinatorial objects: Vector
space partitions of of type , binary MRD
codes of minimum rank distance , and subspace codes with parameters
and .Comment: 31 pages, 9 table
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
Partial Spreads in Random Network Coding
Following the approach by R. K\"otter and F. R. Kschischang, we study network
codes as families of k-dimensional linear subspaces of a vector space F_q^n, q
being a prime power and F_q the finite field with q elements. In particular,
following an idea in finite projective geometry, we introduce a class of
network codes which we call "partial spread codes". Partial spread codes
naturally generalize spread codes. In this paper we provide an easy description
of such codes in terms of matrices, discuss their maximality, and provide an
efficient decoding algorithm
Tables of subspace codes
One of the main problems of subspace coding asks for the maximum possible
cardinality of a subspace code with minimum distance at least over
, where the dimensions of the codewords, which are vector
spaces, are contained in . In the special case of
one speaks of constant dimension codes. Since this (still) emerging
field is very prosperous on the one hand side and there are a lot of
connections to classical objects from Galois geometry it is a bit difficult to
keep or to obtain an overview about the current state of knowledge. To this end
we have implemented an on-line database of the (at least to us) known results
at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated
technical report is to provide a user guide how this technical tool can be used
in research projects and to describe the so far implemented theoretic and
algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot
- …