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arXiv:1106.1414 · IEEE International Symposium on Information Theory (ISIT)

Exact free distance and trapping set growth rates for LDPC convolutional codes

David G. M. Mitchell, Alí Emre Pusane, Michael Lentmaier, Daniel J. Costello

Published Jul 1, 2011• 16 citations
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Abstract

Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC) codes are known to be asymptotically good, in the sense that the minimum free distance grows linearly with the constraint length. In this paper, we use a protograph-based analysis of terminated LDPCC codes to obtain an upper bound on the free distance growth rate of ensembles of periodically time-varying LDPCC codes. This bound is compared to a lower bound and evaluated numerically. It is found that, for a sufficiently large period, the bounds coincide. This approach is then extended to obtain bounds on the trapping set numbers, which define the size of the smallest, non-empty trapping sets, for these asymptotically good, periodically time-varying LDPCC code ensembles.

DOI
10.1109/isit.2011.6033700

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Arvind Sridharan, Dmitri V. Truhachev, Michael Lentmaier +2 more

An ensemble of$(J,K)$-regular low-density parity- check (LDPC) convolutional codes is introduced and existence-type lower bounds on the minimum distance$d _ {\rm L}$of code segments of finite length$L$and on the free distance$d _{\rm free}$are derived. For sufficiently large constraint lengths$\nu$, the distances are shown to grow linearly with$\nu$and the ratio$d_ {\rm L}/\nu$approaches the ratio$d _{ {\rm free}}/\nu$for large$L$. Moreover, the ratio of free distance to constraint length is several times larger than the ratio of minimum distance to block length for Gallager's ensemble of (J,K)-regular LDPC block codes.

IEEE Transactions on Information Theory•Dec 1, 2007•49 citations
Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC) codes are known to be asymptotically good, in the sense that the minimum free distance grows linearly with the constraint length. In this paper, we use a protograph-based analysis of terminated LDPCC codes to obtain an upper bound on the free distance growth rate of ensembles of periodically time-varying LDPCC codes. This bound is compared to a lower bound and evaluated numerically. It is found that, for a sufficiently large period, the bounds coincide. This approach is then extended to obtain bounds on the trapping set numbers, which define the size of the smallest, non-empty trapping sets, for these asymptotically good, periodically time-varying LDPCC code ensembles.
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.
It has been suggested that ldquonear-codewordsrdquo may be a significant factor affecting decoding failures of LDPC codes over the AWGN channel. A near-codeword is a sequence that satisfies almost all of the check equations. These near-codewords can be associated with so-called dasiatrapping setspsila that exist in the Tanner graph of a code. In this paper, we analyse the trapping sets of protograph-based LDPC convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. Further, it has been shown that some ensembles of LDPC convolutional codes are asymptotically good, in the sense that the average free distance grows linearly with constraint length. Here, asymptotic methods are used to calculate a lower bound on the trapping set growth rates for several ensembles of regular asymptotically good protograph-based LDPC convolutional codes. This provides us with an estimate of where the error floor will occur for these codes under iterative message-passing decoding.
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.
It has been suggested that ¿near-codewords¿ may be a significant factor affecting decoding failures of LDPC codes over the AWGN channel. A near-codeword is a sequence that satisfies almost all of the check equations. These near-codewords can be associated with so-called `trapping sets' that exist in the Tanner graph of a code. In this paper, we analyse the trapping sets of protograph-based LDPC convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. Further, it has been shown that some ensembles of LDPC convolutional codes are asymptotically good, in the sense that the average free distance grows linearly with constraint length. Here, asymptotic methods are used to calculate a lower bound on the trapping set growth rates for two ensembles of asymptotically good protograph-based LDPC convolutional codes. This can be used to predict where the error floor will occur for these codes under iterative message-passing decoding.
An ensemble of(J,K)(J,K)(J,K)-regular low-density parity- check (LDPC) convolutional codes is introduced and existence-type lower bounds on the minimum distancedLd _ {\rm L}dL​of code segments of finite lengthLLLand on the free distancedfreed _{\rm free}dfree​are derived. For sufficiently large constraint lengthsν\nuν, the distances are shown to grow linearly withν\nuνand the ratiodL/νd_ {\rm L}/\nudL​/νapproaches the ratiodfree/νd _{ {\rm free}}/\nudfree​/νfor largeLLL. Moreover, the ratio of free distance to constraint length is several times larger than the ratio of minimum distance to block length for Gallager's ensemble of (J,K)-regular LDPC block codes.