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IEEE International Symposium on Information Theory (ISIT)

Design of provably good low-density parity check codes

T. Richardson, Amin Shokrollahi, Rudiger L. Urbanke

Published Nov 7, 2002• 138 citations
Publisher

Abstract

We design sequences of low-density parity check codes that provably perform at rates extremely close to the Shannon capacity. These codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. We further show that under suitable conditions the message densities fulfil a certain symmetry condition which we call the consistency condition and we present a stability condition which is the most powerful tool to date to bound/determine the threshold of a given family of low-density parity check codes.

DOI
10.1109/isit.2000.866497

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We design sequences of low-density parity check codes that provably perform at rates extremely close to the Shannon capacity. These codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. We further show that under suitable conditions the message densities fulfil a certain symmetry condition which we call the consistency condition and we present a stability condition which is the most powerful tool to date to bound/determine the threshold of a given family of low-density parity check codes.
We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds.
Abstract Low-density parity-check (LDPC) codes are among the most efficient error correcting codes in use. This chapter introduces an important family of LDPC ensembles, based on random factor graphs, and studies some of their basic properties. It focuses on performances under optimal decoding, when no constraint is imposed on the computational complexity of the decoding procedure. Bounds in their performances are derived through an analysis of the geometric properties of their codebook. In particular, it shows that appropriately chosen LDPC ensembles allow for communication reliably at rates close to Shannon's capacity.
We construct lattices with high coding gains based on nested low-density parity-check (LDPC) codes by Construction D′.We generalize the LDPC lattices [1] to have irregular degrees and call them irregular LDPC lattices. To construct good irregular LDPC lattices, we optimize the degree distributions by using density evolution and a modified sequential quadratic programming (SQP) and show our optimized irregular LDPC lattice has a threshold 0.48dB from the capacity, which is about 0.6dB better than the regular one in [1]. To construct a Tanner graph corresponding to LDPC lattices, we generalize the progressive-edge growth (PEG) algorithm and show a lower bound on the girth of the graph. Simulation is performed using the sum-product algorithm. Also, we compare the performance of joint decoding and multi-stage decoding [2] and show the advantage of joint decoding.
In this paper, we propose an algorithm to design an irregular low-density parity-check (LDPC) code, with a given degree distribution pair, from a combinatorially constructed regular LDPC code. Richardson et al., (2001) showed that long LDPC codes from irregular bipartite graphs with carefully chosen degree distribution pair performed very close to the Shannon capacity limit. It is known that the cyclic or quasicyclic property of regular LDPC codes, constructed from combinatorial designs Colbourn, J et al., (1996), helps to simplify their encoding procedure and also facilitates a memory-efficient storage of the codes. The proposed algorithm involves splitting columns and rows of a regular LDPC code systematically in order to achieve an irregular code with a given distribution pair. Also, this algorithm is a useful alternative to random generation of irregular codes because it enables to exploit the structural properties of the regular code in efficiently storing the resultant irregular code.
We construct new families of low-density parity-check codes, which we call irregular codes. When decoded using belief propagation, our codes can correct more errors than previously known low-density codes. Our improved performance comes from using codes based on irregular random bipartite graphs, based on the work of Luby et al. (1997). Previously studied low-density codes have been derived from regular bipartite graphs. Initial experimental results for our irregular codes suggest that, with improvements, irregular codes may be able to match turbo code performance.