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arXiv:1303.1858 · IEEE International Symposium on Information Theory (ISIT)

On the minimum distance of generalized spatially coupled LDPC codes

David G. M. Mitchell, Michael Lentmaier, Daniel J. Costello

Published Jul 1, 2013• 23 citations
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Abstract

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.

DOI
10.1109/isit.2013.6620551

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IEEE Information Theory Workshop (ITW)•Apr 1, 2015•1 citation
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.
Families of asymptotically regular LDPC block code ensembles can be formed by terminating (J, K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates and substantially better iterative decoding thresholds than those of (J, K)-regular LDPC block code ensembles, despite the fact that the terminated ensembles are almost regular. Also, by means of an asymptotic weight enumerator analysis, we show that minimum distance grows linearly with block length for all of the ensembles in these families, i.e., the ensembles are asymptotically good. We find that, as the termination length increases, families of ¿asymptotically regular¿ codes with capacity approaching iterative decoding thresholds and declining minimum distance growth rates are obtained, allowing a code designer to trade-off between distance growth rate and threshold. Further, we show that the thresholds and the distance growth rates can be improved by carefully choosing the component protographs used in the code construction.
Spatially coupled low-density parity-check block codes (SC-LDPC-BCs) are a class of LDPC codes in which certain ensembles achieve capacity under iterative belief propagation decoding. Prior work has considered these codes mostly from a protograph perspective. Following this viewpoint, our work presents new protographs for the design of protograph-based SC-LDPC-BCs. A reciprocal channel approximation (RCA) analysis of the terminated protographs yields the asymptotic performance limits of the ensembles in terms of their iterative decoding thresholds. Our protographs possess the properties of linear minimum distance growth rate and excellent thresholds. We consider the following channels in our work: binary-input additive white Gaussian noise channel (BI-AWGNC), binary erasure channel (BEC), and binary symmetric channel (BSC). Our work focuses on obtaining good designs, with two classes of ensembles being asymptotically regular, for all the three channels while minimizing rate loss.
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.
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.
Low-density parity-check (LDPC) convolutional codes have been shown to be capable of achieving capacity-approaching performance with iterative message-passing decoding. In the first part of this paper, using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio, we show that several ensembles of regular and irregular LDPC convolutional codes derived from protograph-based LDPC block codes have the property that the free distance grows linearly with respect to the constraint length, i.e., the ensembles are asymptotically good. In particular, we show that the free distance to constraint length ratio of the LDPC convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding LDPC block code ensembles. A large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal-to-noise ratios under maximum-likelihood decoding. When suboptimal decoding methods are employed, there are many factors that affect the performance of a code. Recently, it has been shown that so-called trapping sets are a significant factor affecting decoding failures of LDPC codes over the additive white Gaussian noise channel with iterative message-passing decoding. In the second part of this paper, we study the trapping sets of the asymptotically good protograph-based LDPC convolutional codes considered earlier. By extending the theory presented in part one and using similar bounding techniques, we show that the size of the smallest non-empty trapping set grows linearly with the constraint length for these ensembles.
In this paper, asymptotic methods are used to form lower and upper bounds on the typical free distance growth rate of ensembles of periodically time-varying protograph-based spatially coupled low-density parity-check (SC-LDPC) codes over GF(q). By evaluating and comparing these bounds, we find that the typical free distance of q-ary SC-LDPC codes increases linearly with constraint length and that the bounds coincide for a sufficiently large period. In particular, we show that the free distance to constraint length ratio of (3, 6)-regular q-ary SC-LDPC code ensembles exceeds the minimum distance to block length ratio of an underlying q-ary LDPC block code (LDPC-BC) ensemble. We also show that, similar to the minimum distance growth rate of the (3, 6)-regular q-ary LDPC-BC ensemble, the free distance growth rate of (3, 6)-regular q-ary SC-LDPC code ensembles increases with the field size q up to a certain point, and then it decreases as q increases further.