7,444 research outputs found
On the Minimum/Stopping Distance of Array Low-Density Parity-Check Codes
In this work, we study the minimum/stopping distance of array low-density
parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code
specified by two integers q and m, where q is an odd prime and m <= q. In the
literature, the minimum/stopping distance of these codes (denoted by d(q,m) and
h(q,m), respectively) has been thoroughly studied for m <= 5. Both exact
results, for small values of q and m, and general (i.e., independent of q)
bounds have been established. For m=6, the best known minimum distance upper
bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002),
is d(q,6) <= 32. In this work, we derive an improved upper bound of d(q,6) <=
20 and a new upper bound d(q,7) <= 24 by using the concept of a template
support matrix of a codeword/stopping set. The bounds are tight with high
probability in the sense that we have not been able to find codewords of
strictly lower weight for several values of q using a minimum distance
probabilistic algorithm. Finally, we provide new specific minimum/stopping
distance results for m <= 7 and low-to-moderate values of q <= 79.Comment: To appear in IEEE Trans. Inf. Theory. The material in this paper was
presented in part at the 2014 IEEE International Symposium on Information
Theory, Honolulu, HI, June/July 201
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
On the Minimum Distance of Array-Based Spatially-Coupled Low-Density Parity-Check Codes
An array low-density parity-check (LDPC) code is a quasi-cyclic LDPC code
specified by two integers and , where is an odd prime and . The exact minimum distance, for small and , has been calculated, and
tight upper bounds on it for have been derived. In this work, we
study the minimum distance of the spatially-coupled version of these codes. In
particular, several tight upper bounds on the optimal minimum distance for
coupling length at least two and , that are independent of and
that are valid for all values of where depends on , are
presented. Furthermore, we show by exhaustive search that by carefully
selecting the edge spreading or unwrapping procedure, the minimum distance
(when is not very large) can be significantly increased, especially for
.Comment: 5 pages. To be presented at the 2015 IEEE International Symposium on
Information Theory, June 14-19, 2015, Hong Kon
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
The Trapping Redundancy of Linear Block Codes
We generalize the notion of the stopping redundancy in order to study the
smallest size of a trapping set in Tanner graphs of linear block codes. In this
context, we introduce the notion of the trapping redundancy of a code, which
quantifies the relationship between the number of redundant rows in any
parity-check matrix of a given code and the size of its smallest trapping set.
Trapping sets with certain parameter sizes are known to cause error-floors in
the performance curves of iterative belief propagation decoders, and it is
therefore important to identify decoding matrices that avoid such sets. Bounds
on the trapping redundancy are obtained using probabilistic and constructive
methods, and the analysis covers both general and elementary trapping sets.
Numerical values for these bounds are computed for the [2640,1320] Margulis
code and the class of projective geometry codes, and compared with some new
code-specific trapping set size estimates.Comment: 12 pages, 4 tables, 1 figure, accepted for publication in IEEE
Transactions on Information Theor
Array Convolutional Low-Density Parity-Check Codes
This paper presents a design technique for obtaining regular time-invariant
low-density parity-check convolutional (RTI-LDPCC) codes with low complexity
and good performance. We start from previous approaches which unwrap a
low-density parity-check (LDPC) block code into an RTI-LDPCC code, and we
obtain a new method to design RTI-LDPCC codes with better performance and
shorter constraint length. Differently from previous techniques, we start the
design from an array LDPC block code. We show that, for codes with high rate, a
performance gain and a reduction in the constraint length are achieved with
respect to previous proposals. Additionally, an increase in the minimum
distance is observed.Comment: 4 pages, 2 figures, accepted for publication in IEEE Communications
Letter
Cyclic lowest density MDS array codes
Three new families of lowest density maximum-distance separable (MDS) array codes are constructed, which are cyclic or quasi-cyclic. In addition to their optimal redundancy (MDS) and optimal update complexity (lowest density), the symmetry offered by the new codes can be utilized for simplified implementation in storage applications. The proof of the code properties has an indirect structure: first MDS codes that are not cyclic are constructed, and then transformed to cyclic codes by a minimum-distance preserving transformation
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