700 research outputs found
On the guaranteed error correction capability of LDPC codes
We investigate the relation between the girth and the guaranteed error
correction capability of -left regular LDPC codes when decoded using
the bit flipping (serial and parallel) algorithms. A lower bound on the number
of variable nodes which expand by a factor of at least is found
based on the Moore bound. An upper bound on the guaranteed correction
capability is established by studying the sizes of smallest possible trapping
sets.Comment: 5 pages, submitted to IEEE International Symposium on Information
Theory (ISIT), 200
On Trapping Sets and Guaranteed Error Correction Capability of LDPC Codes and GLDPC Codes
The relation between the girth and the guaranteed error correction capability
of -left regular LDPC codes when decoded using the bit flipping (serial
and parallel) algorithms is investigated. A lower bound on the size of variable
node sets which expand by a factor of at least is found based on
the Moore bound. An upper bound on the guaranteed error correction capability
is established by studying the sizes of smallest possible trapping sets. The
results are extended to generalized LDPC codes. It is shown that generalized
LDPC codes can correct a linear fraction of errors under the parallel bit
flipping algorithm when the underlying Tanner graph is a good expander. It is
also shown that the bound cannot be improved when is even by studying
a class of trapping sets. A lower bound on the size of variable node sets which
have the required expansion is established.Comment: 17 pages. Submitted to IEEE Transactions on Information Theory. Parts
of this work have been accepted for presentation at the International
Symposium on Information Theory (ISIT'08) and the International Telemetering
Conference (ITC'08
Check-hybrid GLDPC Codes: Systematic Elimination of Trapping Sets and Guaranteed Error Correction Capability
In this paper, we propose a new approach to construct a class of check-hybrid
generalized low-density parity-check (CH-GLDPC) codes which are free of small
trapping sets. The approach is based on converting some selected check nodes
involving a trapping set into super checks corresponding to a 2-error
correcting component code. Specifically, we follow two main purposes to
construct the check-hybrid codes; first, based on the knowledge of the trapping
sets of the global LDPC code, single parity checks are replaced by super checks
to disable the trapping sets. We show that by converting specified single check
nodes, denoted as critical checks, to super checks in a trapping set, the
parallel bit flipping (PBF) decoder corrects the errors on a trapping set and
hence eliminates the trapping set. The second purpose is to minimize the rate
loss caused by replacing the super checks through finding the minimum number of
such critical checks. We also present an algorithm to find critical checks in a
trapping set of column-weight 3 LDPC code and then provide upper bounds on the
minimum number of such critical checks such that the decoder corrects all error
patterns on elementary trapping sets. Moreover, we provide a fixed set for a
class of constructed check-hybrid codes. The guaranteed error correction
capability of the CH-GLDPC codes is also studied. We show that a CH-GLDPC code
in which each variable node is connected to 2 super checks corresponding to a
2-error correcting component code corrects up to 5 errors. The results are also
extended to column-weight 4 LDPC codes. Finally, we investigate the eliminating
of trapping sets of a column-weight 3 LDPC code using the Gallager B decoding
algorithm and generalize the results obtained for the PBF for the Gallager B
decoding algorithm
Two-Bit Bit Flipping Decoding of LDPC Codes
In this paper, we propose a new class of bit flipping algorithms for
low-density parity-check (LDPC) codes over the binary symmetric channel (BSC).
Compared to the regular (parallel or serial) bit flipping algorithms, the
proposed algorithms employ one additional bit at a variable node to represent
its "strength." The introduction of this additional bit increases the
guaranteed error correction capability by a factor of at least 2. An additional
bit can also be employed at a check node to capture information which is
beneficial to decoding. A framework for failure analysis of the proposed
algorithms is described. These algorithms outperform the Gallager A/B algorithm
and the min-sum algorithm at much lower complexity. Concatenation of two-bit
bit flipping algorithms show a potential to approach the performance of belief
propagation (BP) decoding in the error floor region, also at lower complexity.Comment: 6 pages. Submitted to IEEE International Symposium on Information
Theory 201
Construction of Near-Optimum Burst Erasure Correcting Low-Density Parity-Check Codes
In this paper, a simple, general-purpose and effective tool for the design of
low-density parity-check (LDPC) codes for iterative correction of bursts of
erasures is presented. The design method consists in starting from the
parity-check matrix of an LDPC code and developing an optimized parity-check
matrix, with the same performance on the memory-less erasure channel, and
suitable also for the iterative correction of single bursts of erasures. The
parity-check matrix optimization is performed by an algorithm called pivot
searching and swapping (PSS) algorithm, which executes permutations of
carefully chosen columns of the parity-check matrix, after a local analysis of
particular variable nodes called stopping set pivots. This algorithm can be in
principle applied to any LDPC code. If the input parity-check matrix is
designed for achieving good performance on the memory-less erasure channel,
then the code obtained after the application of the PSS algorithm provides good
joint correction of independent erasures and single erasure bursts. Numerical
results are provided in order to show the effectiveness of the PSS algorithm
when applied to different categories of LDPC codes.Comment: 15 pages, 4 figures. IEEE Trans. on Communications, accepted
(submitted in Feb. 2007
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