529 research outputs found
Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three
The concept of group divisible codes, a generalization of group divisible
designs with constant block size, is introduced in this paper. This new class
of codes is shown to be useful in recursive constructions for constant-weight
and constant-composition codes. Large classes of group divisible codes are
constructed which enabled the determination of the sizes of optimal
constant-composition codes of weight three (and specified distance), leaving
only four cases undetermined. Previously, the sizes of constant-composition
codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table
Generalized vector space partitions
A vector space partition in is a set of
subspaces such that every -dimensional subspace of is
contained in exactly one element of . Replacing "every point" by
"every -dimensional subspace", we generalize this notion to vector space
-partitions and study their properties. There is a close connection to
subspace codes and some problems are even interesting and unsolved for the set
case .Comment: 12 pages, typos correcte
Subspace Packings : Constructions and Bounds
The Grassmannian is the set of all -dimensional
subspaces of the vector space . K\"{o}tter and Kschischang
showed that codes in Grassmannian space can be used for error-correction in
random network coding. On the other hand, these codes are -analogs of codes
in the Johnson scheme, i.e., constant dimension codes. These codes of the
Grassmannian also form a family of -analogs of block
designs and they are called subspace designs. In this paper, we examine one of
the last families of -analogs of block designs which was not considered
before. This family, called subspace packings, is the -analog of packings,
and was considered recently for network coding solution for a family of
multicast networks called the generalized combination networks. A subspace
packing - is a set of -subspaces from
such that each -subspace of is
contained in at most elements of . The goal of this work
is to consider the largest size of such subspace packings. We derive a sequence
of lower and upper bounds on the maximum size of such packings, analyse these
bounds, and identify the important problems for further research in this area.Comment: 30 pages, 27 tables, continuation of arXiv:1811.04611, typos
correcte
Load-Balanced Fractional Repetition Codes
We introduce load-balanced fractional repetition (LBFR) codes, which are a
strengthening of fractional repetition (FR) codes. LBFR codes have the
additional property that multiple node failures can be sequentially repaired by
downloading no more than one block from any other node. This allows for better
use of the network, and can additionally reduce the number of disk reads
necessary to repair multiple nodes. We characterize LBFR codes in terms of
their adjacency graphs, and use this characterization to present explicit
constructions LBFR codes with storage capacity comparable existing FR codes.
Surprisingly, in some parameter regimes, our constructions of LBFR codes match
the parameters of the best constructions of FR codes
A family of optimal locally recoverable codes
A code over a finite alphabet is called locally recoverable (LRC) if every
symbol in the encoding is a function of a small number (at most ) other
symbols. We present a family of LRC codes that attain the maximum possible
value of the distance for a given locality parameter and code cardinality. The
codewords are obtained as evaluations of specially constructed polynomials over
a finite field, and reduce to a Reed-Solomon code if the locality parameter
is set to be equal to the code dimension. The size of the code alphabet for
most parameters is only slightly greater than the code length. The recovery
procedure is performed by polynomial interpolation over points. We also
construct codes with several disjoint recovering sets for every symbol. This
construction enables the system to conduct several independent and simultaneous
recovery processes of a specific symbol by accessing different parts of the
codeword. This property enables high availability of frequently accessed data
("hot data").Comment: Minor changes. This is the final published version of the pape
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
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