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IEEE International Symposium on Information Theory (ISIT)

Iterative decoding of generalized low-density parity-check codes

Michael Lentmaier, K.S. Ziganfirov

Published Nov 27, 2002• 43 citations
Publisher

Abstract

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.

DOI
10.1109/isit.1998.708742

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IEEE Communications Letters•Jun 1, 1998•1,201 citations
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.
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.
Introduction In this chapter we introduce low-density parity-check (LDPC) codes, a class of error correction codes proposed by Gallager in his 1962 PhD thesis 12 years after error correction codes were first introduced by Hamming (published in 1950). Both Hamming codes and LDPC codes are block codes: the messages are broken up into blocks to be encoded at the transmitter and similarly decoded as separate blocks at the receiver. While Hamming codes are short and very structured with a known, fixed, error correction ability, LDPC codes are the opposite, usually long and often constructed pseudo-randomly with only a probabilistic notion of their expected error correction performance. The chapter begins by presenting parity bits as a means to detect and, when more than one is employed, to correct errors in digital data. Block error correction codes are described as a linear combination of parity-check equations and thus defined by their parity-check matrix representation. The graphical representation of codes by Tanner graphs is presented and the necessary graph theoretic concepts introduced. In Section 2.4 iterative decoding algorithms are introduced using a hard decision algorithm (bit flipping), so that the topic is developed first without reference to probability theory. Subsequently the sum–product decoding algorithm is presented. This chapter serves as a self-contained introduction to LDPC codes and their decoding. It is intended that the material presented here will enable the reader to implement LDPC encoders and iterative decoders.
Summary form only given. LDPC (low density parity check) codes (Gallager (1963)) are linear parity check codes with a sparse parity check matrix. In our work Gallager's original decoding algorithm is extended to alleviate the problem of channel with finite memory. Two decoding algorithms are derived. The basic algorithm tracks the original derivation of Gallager with the necessary adaptations. This results in a single usage of the BCJR algorithm, whereas the iterative part remains intact.. The second algorithm is a generalization of the first. While the basic algorithm only utilizes the correlation among adjacent digits to achieve an improved estimation of the prior probabilities, the generalized algorithm also exploits the information lying in the parity checks in which the adjacent digits participate. This results in modified rightbound messages. The generalized algorithm is asymptotically optimal in the sense of achieved bit error rate based on the information extracted from a given number of adjacent digits.
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.
Gallager's (1962) low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over GF(q) for q>2, with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1/4 code with bit error probability <10/sup -5/ at E/sub b//N/sub 0/=0.2 dB.