225 research outputs found
Generalized Stability Condition for Generalized and Doubly-Generalized LDPC Codes
In this paper, the stability condition for low-density parity-check (LDPC)
codes on the binary erasure channel (BEC) is extended to generalized LDPC
(GLDPC) codes and doublygeneralized LDPC (D-GLDPC) codes. It is proved that, in
both cases, the stability condition only involves the component codes with
minimum distance 2. The stability condition for GLDPC codes is always expressed
as an upper bound to the decoding threshold. This is not possible for D-GLDPC
codes, unless all the generalized variable nodes have minimum distance at least
3. Furthermore, a condition called derivative matching is defined in the paper.
This condition is sufficient for a GLDPC or DGLDPC code to achieve the
stability condition with equality. If this condition is satisfied, the
threshold of D-GLDPC codes (whose generalized variable nodes have all minimum
distance at least 3) and GLDPC codes can be expressed in closed form.Comment: 5 pages, 2 figures, to appear in Proc. of IEEE ISIT 200
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
Critical Point for Maximum Likelihood Decoding of Linear Block Codes
In this letter, the SNR value at which the error performance curve of a soft
decision maximum likelihood decoder reaches the slope corresponding to the code
minimum distance is determined for a random code. Based on this value, referred
to as the critical point, new insight about soft bounded distance decoding of
random-like codes (and particularly Reed-Solomon codes) is provided.Comment: to appear IEEE Communications Letter
On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes
In this paper, an expression for the asymptotic growth rate of the number of
small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC)
codes is derived. The expression is compact and generalizes existing results
for LDPC and generalized LDPC (GLDPC) codes. Assuming that there exist check
and variable nodes with minimum distance 2, it is shown that the growth rate
depends only on these nodes. An important connection between this new result
and the stability condition of D-GLDPC codes over the BEC is highlighted. Such
a connection, previously observed for LDPC and GLDPC codes, is now extended to
the case of D-GLDPC codes.Comment: 10 pages, 1 figure, presented at the 46th Annual Allerton Conference
on Communication, Control and Computing (this version includes additional
appendix
Spectral Shape of Check-Hybrid GLDPC Codes
This paper analyzes the asymptotic exponent of both the weight spectrum and
the stopping set size spectrum for a class of generalized low-density
parity-check (GLDPC) codes. Specifically, all variable nodes (VNs) are assumed
to have the same degree (regular VN set), while the check node (CN) set is
assumed to be composed of a mixture of different linear block codes (hybrid CN
set). A simple expression for the exponent (which is also referred to as the
growth rate or the spectral shape) is developed. This expression is consistent
with previous results, including the case where the normalized weight or
stopping set size tends to zero. Furthermore, it is shown how certain symmetry
properties of the local weight distribution at the CNs induce a symmetry in the
overall weight spectral shape function.Comment: 6 pages, 3 figures. Presented at the IEEE ICC 2010, Cape Town, South
Africa. A minor typo in equation (9) has been correcte
Growth Rate of the Weight Distribution of Doubly-Generalized LDPC Codes: General Case and Efficient Evaluation
The growth rate of the weight distribution of irregular doubly-generalized
LDPC (D-GLDPC) codes is developed and in the process, a new efficient numerical
technique for its evaluation is presented. The solution involves simultaneous
solution of a 4 x 4 system of polynomial equations. This represents the first
efficient numerical technique for exact evaluation of the growth rate, even for
LDPC codes. The technique is applied to two example D-GLDPC code ensembles.Comment: 6 pages, 1 figure. Proc. IEEE Globecom 2009, Hawaii, USA, November 30
- December 4, 200
Stability of Iterative Decoding of Multi-Edge Type Doubly-Generalized LDPC Codes Over the BEC
Using the EXIT chart approach, a necessary and sufficient condition is
developed for the local stability of iterative decoding of multi-edge type
(MET) doubly-generalized low-density parity-check (D-GLDPC) code ensembles. In
such code ensembles, the use of arbitrary linear block codes as component codes
is combined with the further design of local Tanner graph connectivity through
the use of multiple edge types. The stability condition for these code
ensembles is shown to be succinctly described in terms of the value of the
spectral radius of an appropriately defined polynomial matrix.Comment: 6 pages, 3 figures. Presented at Globecom 2011, Houston, T
Multilevel Block Coded Modulation with Unequal Error Protection
Multilevel block coded modulation (BCM) schemes with unequal error protection (UEP) are investigated. These schemes are based on unconventional set partitions that greatly reduce the error coefficients associated with multi-stage decoding of conventional BCM, at the expense of smaller intra-set distances
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