70,756 research outputs found
New Lower Bounds for Constant Dimension Codes
This paper provides new constructive lower bounds for constant dimension
codes, using different techniques such as Ferrers diagram rank metric codes and
pending blocks. Constructions for two families of parameters of constant
dimension codes are presented. The examples of codes obtained by these
constructions are the largest known constant dimension codes for the given
parameters
Integer linear programming techniques for constant dimension codes and related structures
The lattice of subspaces of a finite dimensional vector space over a finite field is combined with the so-called subspace distance or the injection distance a metric space. A subset of this metric space is called subspace code. If a subspace code contains solely elements, so-called codewords, with equal dimension, it is called constant dimension code, which is abbreviated as CDC. The minimum distance is the smallest pairwise distance of elements of a subspace code. In the case of a CDC, the minimum distance is equivalent to an upper bound on the dimension of the pairwise intersection of any two codewords. Subspace codes play a vital role in the context of random linear network coding, in which data is transmitted from a sender to multiple receivers such that participants of the communication forward random linear combinations of the data. The two main problems of subspace coding are the determination of the cardinality of largest subspace codes and the classification of subspace codes. Using integer linear programming techniques and symmetry, this thesis answers partially the questions above while focusing on CDCs. With the coset construction and the improved linkage construction, we state two general constructions, which improve on the best known lower bound of the cardinality in many cases. A well-structured CDC which is often used as building block for elaborate CDCs is the lifted maximum rank distance code, abbreviated as LMRD. We generalize known upper bounds for CDCs which contain an LMRD, the so-called LMRD bounds. This also provides a new method to extend an LMRD with additional codewords. This technique yields in sporadic cases best lower bounds on the cardinalities of largest CDCs. The improved linkage construction is used to construct an infinite series of CDCs whose cardinalities exceed the LMRD bound. Another construction which contains an LMRD together with an asymptotic analysis in this thesis restricts the ratio between best known lower bound and best known upper bound to at least 61.6% for all parameters. Furthermore, we compare known upper bounds and show new relations between them. This thesis describes also a computer-aided classification of largest binary CDCs in dimension eight, codeword dimension four, and minimum distance six. This is, for non-trivial parameters which in addition do not parametrize the special case of partial spreads, the third set of parameters of which the maximum cardinality is determined and the second set of parameters with a classification of all maximum codes. Provable, some symmetry groups cannot be automorphism groups of large CDCs. Additionally, we provide an algorithm which examines the set of all subgroups of a finite group for a given, with restrictions selectable, property. In the context of CDCs, this algorithm provides on the one hand a list of subgroups, which are eligible for automorphism groups of large codes and on the other hand codes having many symmetries which are found by this method can be enlarged in a postprocessing step. This yields a new largest code in the smallest open case, namely the situation of the binary analogue of the Fano plane
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
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
Bounds on List Decoding of Rank-Metric Codes
So far, there is no polynomial-time list decoding algorithm (beyond half the
minimum distance) for Gabidulin codes. These codes can be seen as the
rank-metric equivalent of Reed--Solomon codes. In this paper, we provide bounds
on the list size of rank-metric codes in order to understand whether
polynomial-time list decoding is possible or whether it works only with
exponential time complexity. Three bounds on the list size are proven. The
first one is a lower exponential bound for Gabidulin codes and shows that for
these codes no polynomial-time list decoding beyond the Johnson radius exists.
Second, an exponential upper bound is derived, which holds for any rank-metric
code of length and minimum rank distance . The third bound proves that
there exists a rank-metric code over \Fqm of length such that the
list size is exponential in the length for any radius greater than half the
minimum rank distance. This implies that there cannot exist a polynomial upper
bound depending only on and similar to the Johnson bound in Hamming
metric. All three rank-metric bounds reveal significant differences to bounds
for codes in Hamming metric.Comment: 10 pages, 2 figures, submitted to IEEE Transactions on Information
Theory, short version presented at ISIT 201
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