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IEEE International Symposium on Information Theory (ISIT)

Check-hybrid GLDPC codes: Systematic elimination of trapping sets by super checks

Vida Ravanmehr, David Declercq, Bane V. Vasic

Published Jun 1, 2014• 3 citations
Publisher

Abstract

In this paper, we propose a new approach to constructing a class of check-hybrid generalized low-density parity-check (GLDPC) codes which are free of small trapping sets. This approach is based on converting selected checks of an LDPC code involving a trapping set to super checks corresponding to a shorter error correcting component code. In particular, we follow two goals in constructing the check-hybrid GLDPC codes: First, the super checks are replaced based on the knowledge of trapping sets of the global LDPC code. We show that by converting only some single checks to super checks the decoder corrects the errors on a trapping set and hence eliminates the trapping set. Second, the number of super checks required for eliminating certain trapping sets is minimized to reduce the rate-loss. We first give an algorithm to find a set of critical checks in a trapping set of an LDPC code and then we provide some upper bounds on the minimum number of critical checks needed to eliminate certain trapping sets in the parity-check matrix of an LDPC code. A possible fixed set for a class of check-hybrid codes is also given.

DOI
10.1109/isit.2014.6874923

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In this paper, we propose a new approach to constructing a class of check-hybrid generalized low-density parity-check (GLDPC) codes which are free of small trapping sets. This approach is based on converting selected checks of an LDPC code involving a trapping set to super checks corresponding to a shorter error correcting component code. In particular, we follow two goals in constructing the check-hybrid GLDPC codes: First, the super checks are replaced based on the knowledge of trapping sets of the global LDPC code. We show that by converting only some single checks to super checks the decoder corrects the errors on a trapping set and hence eliminates the trapping set. Second, the number of super checks required for eliminating certain trapping sets is minimized to reduce the rate-loss. We first give an algorithm to find a set of critical checks in a trapping set of an LDPC code and then we provide some upper bounds on the minimum number of critical checks needed to eliminate certain trapping sets in the parity-check matrix of an LDPC code. A possible fixed set for a class of check-hybrid codes is also given.
In this paper, we propose a new approach to construct a class of check-hybrid generalized low-density parity-check (GLDPC) codes which are free of small trapping sets. The approach is based on converting some selected check nodes involving a trapping set to super checks corresponding to a shorter error-correcting component code. Specifically, we follow two main purposes to construct the check-hybrid codes: First, replacing single parity checks by super checks is done based on the knowledge of the trapping sets of the global LDPC code. We show that by converting some single checks to super checks in a trapping set, the decoder corrects the errors on a trapping set and hence eliminates the trapping set. Second, the rate-loss caused by replacing the super checks is reduced through finding the minimum number of such critical checks. We first present an algorithm to find possible critical checks in a trapping set. We then provide some upper bounds on the minimum number of such critical checks such that the decoder corrects all error patterns on certain trapping sets in the Tanner graph of the global LDPC code. We also provide a potential fixed set for a class of constructed check-hybrid codes.
ABSTRACT 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 2 main purposes to construct the check‐hybrid codes; first, on the basis of the knowledge of trapping sets of an 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 decoder corrects the errors on a trapping set. The second purpose is to minimize the rate loss through finding the minimum number of such critical checks. We also present an algorithm to find critical checks in a trapping set of a column‐weight 3 LDPC code of girth 8 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. 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 of girth 6. Finally, we investigate eliminating of trapping sets of a column‐weight 3 LDPC code of girth 8 using the Gallager B decoding algorithm.
Due to the more complex constraints, generalized low-density parity-check (GLDPC) codes can achieve better error performance but require much higher decoding complexities compared with the standard LDPC codes. In this paper, single-parity-check product-codes (SPC-PCs) are considered as the constituent codes in the super check nodes of a GLDPC code. Moreover, turbo iterations are used in decoding the SPC-PCs. The error performance and the decoder complexity of the proposed GLDPC code are compared with other LDPC code and GLDPC code.
Generalized low-density parity-check (GLDPC) codes have emerged as a promising class of error-correcting codes that extend the capabilities of traditional LDPC codes by incorporating more powerful component codes. This paper investigates an approach to construct long GLDPC codes using short component codes. In this scheme, the BCJR algorithm is used as a component decoder and integrated into the belief propagation algorithm. Simulation results demonstrate that constructions incorporating quasi-cyclic LDPC codes yield the best performance. Compared to the conventional decoding approach for LDPC codes, the proposed generalized constructions demonstrated better bit and frame error rate performance. This approach is promising for communication systems that require low latency. A comparison is also provided with LDPC codes constructed using the Progressive Edge-Growth (PEG) algorithm.
A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.
This paper is concerned with the construction of low-density parity-check (LDPC) codes with low error floors. Two main contributions are made. First, a new class of structured LDPC codes is introduced. The parity check matrices of these codes are arrays of permutation matrices which are obtained from Latin squares and form a finite field under some matrix operations. Second, a method to construct LDPC codes with low error floors on the binary symmetric channel (BSC) is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum product algorithm (SPA) by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets.