330 research outputs found
Codes and Designs Related to Lifted MRD Codes
Lifted maximum rank distance (MRD) codes, which are constant dimension codes,
are considered. It is shown that a lifted MRD code can be represented in such a
way that it forms a block design known as a transversal design. A slightly
different representation of this design makes it similar to a analog of a
transversal design. The structure of these designs is used to obtain upper
bounds on the sizes of constant dimension codes which contain a lifted MRD
code. Codes which attain these bounds are constructed. These codes are the
largest known codes for the given parameters. These transversal designs can be
also used to derive a new family of linear codes in the Hamming space. Bounds
on the minimum distance and the dimension of such codes are given.Comment: Submitted to IEEE Transactions on Information Theory. The material in
this paper was presented in part in the 2011 IEEE International Symposium on
Information Theory, Saint Petersburg, Russia, August 201
Covering of Subspaces by Subspaces
Lower and upper bounds on the size of a covering of subspaces in the
Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph
\cG_q(n,k), , are discussed. The problem is of interest from four
points of view: coding theory, combinatorial designs, -analogs, and
projective geometry. In particular we examine coverings based on lifted maximum
rank distance codes, combined with spreads and a recursive construction. New
constructions are given for with or . We discuss the density
for some of these coverings. Tables for the best known coverings, for and
, are presented. We present some questions concerning
possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352
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
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
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
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