83 research outputs found
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
The interplay of different metrics for the construction of constant dimension codes
A basic problem for constant dimension codes is to determine the maximum
possible size of a set of -dimensional subspaces in
, called codewords, such that the subspace distance satisfies
for all pairs of different codewords ,
. Constant dimension codes have applications in e.g.\ random linear network
coding, cryptography, and distributed storage. Bounds for are the
topic of many recent research papers. Providing a general framework we survey
many of the latest constructions and show up the potential for further
improvements. As examples we give improved constructions for the cases
, , , and . We also derive
general upper bounds for subcodes arising in those constructions.Comment: 19 pages; typos correcte
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Contemporary Coding Theory
Coding Theory naturally lies at the intersection of a large number
of disciplines in pure and applied mathematics. A multitude of
methods and means has been designed to construct, analyze, and
decode the resulting codes for communication. This has suggested to
bring together researchers in a variety of disciplines within
Mathematics, Computer Science, and Electrical Engineering, in order
to cross-fertilize generation of new ideas and force global
advancement of the field. Areas to be covered are Network Coding,
Subspace Designs, General Algebraic Coding Theory, Distributed
Storage and Private Information Retrieval (PIR), as well as
Code-Based Cryptography
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
Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4
It is shown that the maximum size of a binary subspace code of packet length
, minimum subspace distance , and constant dimension is ;
in Finite Geometry terms, the maximum number of planes in
mutually intersecting in at most a point is .
Optimal binary subspace codes are classified into
isomorphism types, and a computer-free construction of one isomorphism type is
provided. The construction uses both geometry and finite fields theory and
generalizes to any , yielding a new family of -ary
subspace codes
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