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IEEE International Symposium on Information Theory (ISIT)

Improving spatially coupled LDPC codes by connecting chains

Dmitri V. Truhachev, David G. M. Mitchell, Michael Lentmaier, Daniel J. Costello

Published Jul 1, 2012• 10 citations
Publisher

Abstract

In this paper, we study ensembles of connected spatially coupled low-density parity-check codes (SC-LDPCCs), i.e., ensembles described by graphs in which regular SC-LDPCC chains of various lengths serve as edges. We show that, by carefully connecting individual SC-LDPCC chains, we obtain LDPC code ensembles with improved iterative decoding thresholds compared to those of a single coupled chain, in addition to reducing the decoding complexity required to achieve a specific bit error probability. Moreover, we show that, like the component SC-LDPCC chains, the proposed constructions have a typical minimum distance that grows linearly with block length.

DOI
10.1109/isit.2012.6284232

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IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2015•2 citations
In this paper, we study ensembles of connected spatially coupled low-density parity-check codes (SC-LDPCCs), i.e., ensembles described by graphs in which regular SC-LDPCC chains of various lengths serve as edges. We show that, by carefully connecting individual SC-LDPCC chains, we obtain LDPC code ensembles with improved iterative decoding thresholds compared to those of a single coupled chain, in addition to reducing the decoding complexity required to achieve a specific bit error probability. Moreover, we show that, like the component SC-LDPCC chains, the proposed constructions have a typical minimum distance that grows linearly with block length.
Codes constructed from connected spatially coupled low-density parity-check code (SC-LDPCC) chains are proposed and analyzed. It is demonstrated that connecting coupled chains results in improved iterative decoding performance. The constructed protograph ensembles have better iterative decoding thresholds compared to an individual SC-LDPCC chain and require less computational complexity per bit when operating in the near-threshold region. In addition, it is shown that the proposed constructions are asymptotically good in terms of minimum distance.
The finite length performance of codes on graphs constructed by connecting spatially coupled low-density parity-check (SC-LDPC) code chains is analyzed. Successive (peeling) decoding is considered for the binary erasure channel (BEC). The evolution of the undecoded portion of the bipartite graph remaining after each iteration is analyzed as a dynamical system. It is shown that, in addition to superior iterative decoding thresholds, connected chain ensembles have better performance than single chain ensembles of the same rate and length.
In this paper, we construct protograph-based spatially coupled low-density parity-check (LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L , we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L . We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L , the BP thresholds on both the binary erasure channel and the binary-input additive white Gaussian noise channel saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output channels with low-complexity BP decoding.
Families of generalized spatially-coupled low-density parity-check (GSC-LDPC) code ensembles can be formed by terminating protograph-based generalized LDPC convolutional (GLDPCC) codes. It has previously been shown that ensembles of GSC-LDPC codes constructed from a protograph have better iterative decoding thresholds than their block code counterparts, and that, for large termination lengths, their thresholds coincide with the maximum a-posteriori (MAP) decoding threshold of the underlying generalized LDPC block code ensemble. Here we show that, in addition to their excellent iterative decoding thresholds, ensembles of GSC-LDPC codes are asymptotically good and have large minimum distance growth rates.
The newest family of low-density parity-check (LDPC) codes, spatially-coupled (SC) codes, is shown to have several desirable characteristics including low implementation complexity and close-to-optimal performance over a range of channels. Furthermore, because of their ribbon-shaped parity-check matrices, window decoding can be used to decode these codes, which leads to low-delay implementations. Researchers have focused on asymptotically regular SC code ensembles and have examined several aspects of the code construction processes. In this paper, we concentrate on irregular SC code ensembles. We evaluate their decoding thresholds over the binary erasure channel and show that their performance is better than their regular SC counterparts. It is also shown that the gap between asymptotic coding thresholds of irregular SC ensembles and the fundamental Shannon limit gets negligibly small. For the sake of a better comparison, we have also evaluated the finite-length error performance of selected regular and irregular SC codes over the additive white Gaussian channel and it is also observed that finite-length error performance of these irregular SC codes outperforms regular SC codes. To further improve the error performance of these codes and to lower the possible error floors, progressive edge growth algorithm has also been considered in the finite-length performance analysis.
The finite-length performance of multi-edge spatially coupled low-density parity-check (SC-LDPC) codes over the binary erasure channel (BEC) is analyzed. Existing scaling laws are extended to arbitrary protograph base matrices that include puncturing patterns and multiple edges between nodes. A regular protograph-based SC-LDPC construction based on the (4; 8)-regular LDPC block code works well in the waterfall region compared to more involved rate-1/2 structures proposed to improve the threshold to minimum distance trade-off. Scaling laws are also used for code design and to estimate the block length of a given SC-LDPC code ensemble to match the performance of some other code. Estimates on the performance degradation are developed if the chain length varies.