3,694 research outputs found
Weight Distributions of Regular Low-Density Parity-Check Codes over Finite Fields
The average weight distribution of a regular low-density parity-check (LDPC)
code ensemble over a finite field is thoroughly analyzed. In particular, a
precise asymptotic approximation of the average weight distribution is derived
for the small-weight case, and a series of fundamental qualitative properties
of the asymptotic growth rate of the average weight distribution are proved.
Based on this analysis, a general result, including all previous results as
special cases, is established for the minimum distance of individual codes in a
regular LDPC code ensemble.Comment: 15 pages, 5 figures, accepted for publication in IEEE Transactions on
Information Theory, July 201
Error-Correction Performance of Regular Ring-Linear LDPC Codes over Lee Channels
Most low-density parity-check (LDPC) code constructions are considered over finite fields. In this work, we focus on regular LDPC codes over integer residue rings and analyze their performance with respect to the Lee metric. Their error-correction performance is studied over two channel models, in the Lee metric. The first channel model is a discrete memoryless channel, whereas in the second channel model an error vector is drawn uniformly at random from all vectors of a fixed Lee weight. It is known that the two channel laws coincide in the asymptotic regime, meaning that their marginal distributions match. For both channel models, we derive upper bounds on the block error probability in terms of a random coding union bound as well as sphere packing bounds that make use of the marginal distribution of the considered channels. We estimate the decoding error probability of regular LDPC code ensembles over the channels using the marginal distribution and determining the expected Lee weight distribution of a random LDPC code over a finite integer ring. By means of density evolution and finite-length simulations, we estimate the error-correction performance of selected LDPC code ensembles under belief propagation decoding and a low-complexity symbol message passing decoding algorithm and compare the performances
Second-Order Weight Distributions
A fundamental property of codes, the second-order weight distribution, is
proposed to solve the problems such as computing second moments of weight
distributions of linear code ensembles. A series of results, parallel to those
for weight distributions, is established for second-order weight distributions.
In particular, an analogue of MacWilliams identities is proved. The
second-order weight distributions of regular LDPC code ensembles are then
computed. As easy consequences, the second moments of weight distributions of
regular LDPC code ensembles are obtained. Furthermore, the application of
second-order weight distributions in random coding approach is discussed. The
second-order weight distributions of the ensembles generated by a so-called
2-good random generator or parity-check matrix are computed, where a 2-good
random matrix is a kind of generalization of the uniformly distributed random
matrix over a finite filed and is very useful for solving problems that involve
pairwise or triple-wise properties of sequences. It is shown that the 2-good
property is reflected in the second-order weight distribution, which thus plays
a fundamental role in some well-known problems in coding theory and
combinatorics. An example of linear intersecting codes is finally provided to
illustrate this fact.Comment: 10 pages, accepted for publication in IEEE Transactions on
Information Theory, May 201
Circulant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes
This paper consists of three parts. The first part presents a large class of
new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose
parity-check matrices are constructed based on cyclic subgroups of finite
fields. Experimental results show that the codes constructed perform well over
the binary-input AWGN channel with iterative decoding using the sum-product
algorithm (SPA). The second part analyzes the ranks of the parity-check
matrices of codes constructed based on finite fields with characteristic of 2
and gives combinatorial expressions for these ranks. The third part identifies
a subclass of constructed QC-LDPC codes that have large minimum distances.
Decoding of codes in this subclass with the SPA converges very fast.Comment: 26 pages, 6 figures, submitted to IEEE Transaction on Communication
Low-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes
This paper is concerned with general analysis on the rank and row-redundancy
of an array of circulants whose null space defines a QC-LDPC code. Based on the
Fourier transform and the properties of conjugacy classes and Hadamard products
of matrices, we derive tight upper bounds on rank and row-redundancy for
general array of circulants, which make it possible to consider row-redundancy
in constructions of QC-LDPC codes to achieve better performance. We further
investigate the rank of two types of construction of QC-LDPC codes:
constructions based on Vandermonde Matrices and Latin Squares and give
combinatorial expression of the exact rank in some specific cases, which
demonstrates the tightness of the bound we derive. Moreover, several types of
new construction of QC-LDPC codes with large row-redundancy are presented and
analyzed.Comment: arXiv admin note: text overlap with arXiv:1004.118
Multiplicatively Repeated Non-Binary LDPC Codes
We propose non-binary LDPC codes concatenated with multiplicative repetition
codes. By multiplicatively repeating the (2,3)-regular non-binary LDPC mother
code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9,
1/12,... Surprisingly, such simple low-rate non-binary LDPC codes outperform
the best low-rate binary LDPC codes so far. Moreover, we propose the decoding
algorithm for the proposed codes, which can be decoded with almost the same
computational complexity as that of the mother code.Comment: To appear in IEEE Transactions on Information Theor
Statistical mechanics of error exponents for error-correcting codes
Error exponents characterize the exponential decay, when increasing message
length, of the probability of error of many error-correcting codes. To tackle
the long standing problem of computing them exactly, we introduce a general,
thermodynamic, formalism that we illustrate with maximum-likelihood decoding of
low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and
the binary symmetric channel (BSC). In this formalism, we apply the cavity
method for large deviations to derive expressions for both the average and
typical error exponents, which differ by the procedure used to select the codes
from specified ensembles. When decreasing the noise intensity, we find that two
phase transitions take place, at two different levels: a glass to ferromagnetic
transition in the space of codewords, and a paramagnetic to glass transition in
the space of codes.Comment: 32 pages, 13 figure
The Dynamic Phase Transition for Decoding Algorithms
The state-of-the-art error correcting codes are based on large random
constructions (random graphs, random permutations, ...) and are decoded by
linear-time iterative algorithms. Because of these features, they are
remarkable examples of diluted mean-field spin glasses, both from the static
and from the dynamic points of view. We analyze the behavior of decoding
algorithms using the mapping onto statistical-physics models. This allows to
understand the intrinsic (i.e. algorithm independent) features of this
behavior.Comment: 40 pages, 29 eps figure
Bit-Metric Decoding of Non-Binary LDPC Codes with Probabilistic Amplitude Shaping
A new approach for combining non-binary low-density parity-check (NB-LDPC)
codes with higher-order modulation and probabilistic amplitude shaping (PAS) is
presented. Instead of symbol-metric decoding (SMD), a bit-metric decoder (BMD)
is used so that matching the field order of the non-binary code to the
constellation size is not needed, which increases the flexibility of the coding
scheme. Information rates, density evolution thresholds and finite-length
simulations show that the flexibility comes at no loss of performance if PAS is
used.Comment: Accepted for IEEE Communication Letter
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