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International Symposium on Information Theory and its Applications

Analysis of Generalized LDPC Codes with Random Component Codes for the Binary Erasure Channel

Enrico Paolini, M.P.C. Fossorier, Marco Chiani

Published Jan 1, 2006• 12 citations
OpenAlex

Abstract

In this paper, a method for the asymptotic analysis of generalized low-density parity-check (GLDPC) codes on the binary erasure channel (BEC) is proposed. The considered GLDPC codes have block linear codes as check nodes. Instead of considering specific check com-ponent codes, like Hamming or BCH codes, random codes are considered, and a technique is developed for obtaining the expected check EXIT function for the overall GLDPC code. Each check component code is supposed to belong to an expurgated ensemble. Some GLDPC thresholds obtained by this technique are com-pared with those of GLDPC codes, with the same dis-tribution and component codes lengths, using specific codes. Results obtained by combining our analysis with differential evolution tool are also presented. 1.

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In this paper, a method for the asymptotic analysis of generalized low-density parity-check (GLDPC) codes on the binary erasure channel (BEC) is proposed. The considered GLDPC codes have block linear codes as check nodes. Instead of considering specific check com-ponent codes, like Hamming or BCH codes, random codes are considered, and a technique is developed for obtaining the expected check EXIT function for the overall GLDPC code. Each check component code is supposed to belong to an expurgated ensemble. Some GLDPC thresholds obtained by this technique are com-pared with those of GLDPC codes, with the same dis-tribution and component codes lengths, using specific codes. Results obtained by combining our analysis with differential evolution tool are also presented. 1.
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.
Ensemble design of low-density parity-check (LDPC) codes and their generalizations is usually performed via numerical optimization techniques, such as differential evolution, in which a threshold analysis tool is always necessary. Threshold analysis of unstructured doubly-generalized LDPC (D-GLDPC) code ensembles over the binary erasure channel (BEC) can be performed via extrinsic information transfer (EXIT) chart, exploiting the information functions and split information functions of the check and variable component codes, respectively. In this paper, multi-type information functions of linear block codes are introduced as an extension of the concept of information functions, when the bit positions are assumed to be associated with different types. It is shown how multi-type information functions (together with their split counterparts) can be exploited within an EXIT analysis approach to perform threshold analysis over the BEC of multi-edge type D-GLDPC code ensembles. The proposed technique for threshold analysis captures D-GLDPC codes based on protographs as a special case.
In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over the binary erasure channel (BEC). The main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of LDPC codes when decoded iteratively. We also give expressions for upper bounds on the average bit and block erasure probability for regular LDPC ensembles and the standard random ensemble under maximum-likelihood (ML) decoding. Finally, we present what we consider to be the most important open problems in this area.
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. The performance of the proposed scheme is investigated on a BEC and BSC for infinite and finite code lengths. The analysis shows that the performance of the scheme on a BEC is poor compared to that of LDPC codes. However, the performance on a BSC is competitive to that of LDPC codes. Furthermore, results of the finite length analysis on a BEC can be used under certain conditions as a tight lower bound on the performance of the scheme on a BSC.
The design of low-density parity-check (LDPC) code ensembles optimized for a finite number of decoder iterations is investigated. Our approach employs EXIT chart analysis and differential evolution to design such ensembles for the binary erasure channel and additive white Gaussian noise channel. The error rates of codes optimized for various numbers of decoder iterations are compared and it is seen that in the cases considered, the best performance for a given number of decoder iterations is achieved by codes which are optimized for this particular number. The design of generalized LDPC (GLDPC) codes is also considered, showing that these structures can offer better performance than LDPC codes for low-iteration-number designs. Finally, it is illustrated that LDPC codes which are optimized for a small number of iterations exhibit significant deviations in terms of degree distribution and weight enumerators with respect to LDPC codes returned by more conventional design tools.
The design of low-density parity-check (LDPC) code ensembles optimized for a finite number of decoder iterations is investigated. Our approach employs EXIT chart analysis and differential evolution to design such ensembles for the binary erasure channel and additive white Gaussian noise channel. The error rates of codes optimized for various numbers of decoder iterations are compared and it is seen that in the cases considered, the best performance for a given number of decoder iterations is achieved by codes which are optimized for this particular number. The design of generalized LDPC (GLDPC) codes is also considered, showing that these structures can offer better performance than LDPC codes for low-iteration-number designs. Finally, it is illustrated that LDPC codes which are optimized for a small number of iterations exhibit significant deviations in terms of degree distribution and weight enumerators with respect to LDPC codes returned by more conventional design tools.