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IEEE Communications Letters

On generalized low-density parity-check codes based on Hamming component codes

Michael Lentmaier, Kamil Sh. Zigangirov

Published Aug 1, 1999• 190 citations
Publisher

Abstract

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.

DOI
10.1109/4234.781010

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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.
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.
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.
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 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.
In this paper we construct three types of low-density parity-check codes with column weight j = 3 based on geometries in graphical models. Low-density parity-check codes with j > 2 are desired because their minimum distance improves linearly with the code block length n. The codes we present here have girth 8 and girth 10. All codes are regular and well-structured. These codes have flexible block lengths and code rates, and may be used in the area of communications and data storage. Our simulation results show that they have better bit-error-rate decoding performance and lower error floors in additive white Gaussian noise channels than randomly constructed low-density parity-check codes.
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.