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IEEE AFRICON

Generalized irregular low-density (Tanner) codes based on Hamming component codes

Telex M. N. Ngatched, Fambirai Takawira

Published Aug 27, 2003• 2 citations
Publisher

Abstract

In this paper we introduce the construction of new families of error-correcting codes based on Irregular low-density parity-check codes, which we called generalized irregular low density (GILD) codes, where as component codes, single-error correcting Hamming codes are used instead of single-error detecting parity-check codes. The decoding of GILD Is based on simple and fast SISO (soft input-soft output) decoding of Hamming component codes. Simulation results over an AWGN channel indicate that excellent performances can be achieved.

DOI
10.1109/afrcon.2002.1146832

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In this paper we introduce the construction of new families of error-correcting codes based on Irregular low-density parity-check codes, which we called generalized irregular low density (GILD) codes, where as component codes, single-error correcting Hamming codes are used instead of single-error detecting parity-check codes. The decoding of GILD Is based on simple and fast SISO (soft input-soft output) decoding of Hamming component codes. Simulation results over an AWGN channel indicate that excellent performances can be achieved.
Low-density parity-check (LDPC) codes form a Shannon limit approaching class of linear block codes. With iterative decoding based on their Tanner graphs, they can achieve outstanding performance. Since their rediscovery in late 1990's, the design, construction, and decoding of LDPC codes as well as their generalization have become one of the focal research points. This thesis takes a few more steps in these directions. The first significant contribution of this thesis is the introduction of a new class of codes called Generalized Irregular Low-Density (GILD) parity-check codes, which are adapted from the previously known class of Generalized Low-Density (GLD) codes. GILD codes are generalization of irregular LDPC codes, and are shown to outperform GLD codes. In addition, GILD codes have a significant advantage over GLD codes in terms of encoding and decoding complexity. They are also able to match and even beat LDPC codes for small block lengths. The second significant contribution of this thesis is the proposition of several decoding algorithms. Two new decoding algolithms for LDPC codes are introduced. In principle and complexity these algorithms can be grouped with bit flipping algorithms. Two soft-input soft-output (SISO) decoding algorithms for linear block codes are also proposed. The first algorithm is based on Maximum a Posteriori Probability (MAP) decoding of low-weight subtrellis centered around a generated candidate codeword. The second algorithm modifies and utilizes the improved Kaneko's decoding algorithm for soft-input hard-output decoding. These hard outputs are converted to soft-decisions using reliability calculations. Simulation results indicate that the proposed algorithms provide a significant improvement in error performance over Chase-based algorithm and achieve practically optimal performance with a significant reduction in decoding complexity. An analytical expression for the union bound on the bit error probability of linear codes on the Gilbert-Elliott (GE) channel model is also derived. This analytical result is shown to be accurate in establishing the decoder performance in the range where obtaining sufficient data from simulation is impractical.
A generalization of Gallager's low-density parity-check codes is introduced, where as component codes single-error correcting Hamming codes are used instead of single-error detecting parity-check codes. Low-density (LD) parity-check codes were first introduced by Gallager in 1963. These codes are, in combination with iterative decoding, very promising for achieving low error probabilities at a reasonable cost. Results of computer simulations for long LD codes show, that a performance close to the Shannon capacity limit can be achieved. In this work, we consider a generalized version of low-density parity-check codes, where the decoding procedure can be based on the decoding of Hamming component codes.
A class of pseudo-random compound error-correcting codes, called generalized low density (GLD) parity-check codes, has been proposed recently. As a generalization of Gallager's low-density parity check (LPDC) codes, GLD codes are also asymptotically good in the sense of minimum distance criterion and can be effectively decoded based on iterative soft-input soft-output (SISO) decoding of individual constituent codes. The code performance and decoding complexity of GLD codes are heavily dependent on the employed SISO decoding algorithm. In this paper, we present an efficient SISO decoding algorithm for GLD codes. The proposed algorithm utilizes Kaneko's decoding algorithm for soft-input hard-output decoding. These hard outputs are converted to soft-decision using reliability calculations. The algorithm proposed here presents a major advantage over existing decoding algorithms for GLD codes by being applicable even when long and powerful component codes are used. The complexity of the proposed algorithm is less than that of trellis-based decoding algorithms with a performance degradation of only 0.3 dB.
In this paper we investigate a generalization of Gallager's (1963) low-density (LD) parity-check codes, where as component codes single error correcting Hamming codes are used instead of single error detecting parity-check codes. It is proved that there exist such generalized low-density (GLD) codes for which the minimum distance is growing linearly with the block length, and a lower bound of the minimum distance is given. We also study iterative decoding of GLD codes for the communication over an additive white Gaussian noise channel. The performance in terms of the bit error rate, obtained by computer simulations, is presented for GLD codes of different lengths.
We build a class of pseudo-random error correcting codes, called generalized low density codes (GLD), from the intersection of two interleaved block codes. GLD code performance approaches the channel capacity limit and the GLD decoder is based on simple and fast SISO (soft input-soft output) decoders of smaller block codes. GLD codes are a special case of Tanner codes and a generalization of Gallager's LDPC codes. It is also proved by an ensemble performance argument that these codes are asymptotically good in the sense of the minimum distance criterion. The flexibility in selecting the parameters of GLD codes makes them suitable for small and large block length forward error correcting schemes.
We construct new families of error-correcting codes based on Gallager's (1973) low-density parity-check codes. We improve on Gallager's results by introducing irregular parity-check matrices and a new rigorous analysis of hard-decision decoding of these codes. We also provide efficient methods for finding good irregular structures for such decoding algorithms. Our rigorous analysis based on martingales, our methodology for constructing good irregular codes, and the demonstration that irregular structure improves performance constitute key points of our contribution. We also consider irregular codes under belief propagation. We report the results of experiments testing the efficacy of irregular codes on both binary-symmetric and Gaussian channels. For example, using belief propagation, for rate 1/4 codes on 16000 bits over a binary-symmetric channel, previous low-density parity-check codes can correct up to approximately 16% errors, while our codes correct over 17%. In some cases our results come very close to reported results for turbo codes, suggesting that variations of irregular low density parity-check codes may be able to match or beat turbo code performance.