18 research outputs found
Iterative Effect on LDPC Code Performance
The introduction of the turbo-codes in the early 90’s and, more generally of the iterative principle applied to the treatment of the signal, revolutionized the manner of improving a numerical communication system. This notable projection allowed the rediscovery of the error correcting codes invented by R. Gallager in 1963, called Low-Density Parity-Check codes (LDPC). These codes will be studied in this paper and more particularly the regular LDPC codes and its iterative effect on the performances of these codes on a gaussian transmission channel
Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions
This paper examines the construction of low-density parity-check (LDPC) codes
from transversal designs based on sets of mutually orthogonal Latin squares
(MOLS). By transferring the concept of configurations in combinatorial designs
to the level of Latin squares, we thoroughly investigate the occurrence and
avoidance of stopping sets for the arising codes. Stopping sets are known to
determine the decoding performance over the binary erasure channel and should
be avoided for small sizes. Based on large sets of simple-structured MOLS, we
derive powerful constraints for the choice of suitable subsets, leading to
improved stopping set distributions for the corresponding codes. We focus on
LDPC codes with column weight 4, but the results are also applicable for the
construction of codes with higher column weights. Finally, we show that a
subclass of the presented codes has quasi-cyclic structure which allows
low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications
Shortened Array Codes of Large Girth
One approach to designing structured low-density parity-check (LDPC) codes
with large girth is to shorten codes with small girth in such a manner that the
deleted columns of the parity-check matrix contain all the variables involved
in short cycles. This approach is especially effective if the parity-check
matrix of a code is a matrix composed of blocks of circulant permutation
matrices, as is the case for the class of codes known as array codes. We show
how to shorten array codes by deleting certain columns of their parity-check
matrices so as to increase their girth. The shortening approach is based on the
observation that for array codes, and in fact for a slightly more general class
of LDPC codes, the cycles in the corresponding Tanner graph are governed by
certain homogeneous linear equations with integer coefficients. Consequently,
we can selectively eliminate cycles from an array code by only retaining those
columns from the parity-check matrix of the original code that are indexed by
integer sequences that do not contain solutions to the equations governing
those cycles. We provide Ramsey-theoretic estimates for the maximum number of
columns that can be retained from the original parity-check matrix with the
property that the sequence of their indices avoid solutions to various types of
cycle-governing equations. This translates to estimates of the rate penalty
incurred in shortening a code to eliminate cycles. Simulation results show that
for the codes considered, shortening them to increase the girth can lead to
significant gains in signal-to-noise ratio in the case of communication over an
additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information
Theory, Aug 200
Construction of LDPC convolutional codes via difference triangle sets
In this paper, a construction of LDPC convolutional codes over
arbitrary finite fields, which generalizes the work of Robinson and Bernstein
and the later work of Tong is provided. The sets of integers forming a
-(weak) difference triangle set are used as supports of some columns of
the sliding parity-check matrix of an convolutional code, where
, . The parameters of the convolutional code are related
to the parameters of the underlying difference triangle set. In particular, a
relation between the free distance of the code and is established as well
as a relation between the degree of the code and the scope of the difference
triangle set. Moreover, we show that some conditions on the weak difference
triangle set ensure that the Tanner graph associated to the sliding
parity-check matrix of the convolutional code is free from -cycles not
satisfying the full rank condition over any finite field. Finally, we relax
these conditions and provide a lower bound on the field size, depending on the
parity of , that is sufficient to still avoid -cycles. This is
important for improving the performance of a code and avoiding the presence of
low-weight codewords and absorbing sets.Comment: 22 pages, Extended version of arXiv:2001.0796
Moderate-density parity-check codes from projective bundles
New constructions for moderate-density parity-check (MDPC) codes using finite geometry are proposed. We design a parity-check matrix for the main family of binary codes as the concatenation of two matrices: the incidence matrix between points and lines of the Desarguesian projective plane and the incidence matrix between points and ovals of a projective bundle. A projective bundle is a special collection of ovals which pairwise meet in a unique point. We determine the minimum distance and the dimension of these codes, and we show that they have a natural quasi-cyclic structure. We consider alternative constructions based on an incidence matrix of a Desarguesian projective plane and compare their error-correction performance with regards to a modification of Gallager’s bit-flipping decoding algorithm. In this setting, our codes have the best possible error-correction performance after one round of bit-flipping decoding given the parameters of the code’s parity-check matrix
Searching for Voltage Graph-Based LDPC Tailbiting Codes with Large Girth
The relation between parity-check matrices of quasi-cyclic (QC) low-density
parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports
searching for powerful LDPC block codes. Using the principle of tailbiting,
compact representations of bipartite graphs based on convolutional codes can be
found.
Bounds on the girth and the minimum distance of LDPC block codes constructed
in such a way are discussed. Algorithms for searching iteratively for LDPC
block codes with large girth and for determining their minimum distance are
presented. Constructions based on all-ones matrices, Steiner Triple Systems,
and QC block codes are introduced. Finally, new QC regular LDPC block codes
with girth up to 24 are given.Comment: Submitted to IEEE Transactions on Information Theory in February 201
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
Regular low-density parity-check codes from combinatorial designs
Analytically constructed LDPC codes comprise only a very small subset of possible codes and as a result LDPC codes are still, for the most part, constructed randomly. This paper extends the class of LDPC codes that can be systematically generated by presenting a construction method for regular LDPC codes based on combinatorial designs known as Kirkman triple systems. We construct (3, ρ)-regular codes whose Tanner graph is free of 4-cycles for any integer ρ, and examine girth and minimum distance properties of several classes of LDPC codes obtained from combinatorial designs