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IEEE International Symposium on Information Theory (ISIT)

Stopping set distributions of two-edge type LDPC code ensembles

Ryoji Ikegaya, Kazuhiko Kasai, Yuji Shimoyama, T. Shibuya, K. Sakaniwa

Published Jan 1, 2005• 3 citations
Publisher

Abstract

In this paper, we explicitly formulate the average stopping set distributions and their asymptotic exponents of two instances of two-edge type LDPC code ensembles. Further we investigate the relation between the asymptotic exponents of those two code ensembles
DOI
10.1109/isit.2005.1523485

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In this paper, we study the asymptotic behavior of two quantities: the distribution of stopping sets, and the stopping number s/sup */-the size of the smallest nonempty stopping set in a code and use it to bound the error probability of LDPC codes with large blocklength. We derive an expression for the normalized average stopping set distribution of regular and irregular LDPC codes and use it to obtain a critical fraction of the block length above which codes have exponentially many stopping sets of that size.
Stopping sets determine the performance of low-density parity-check (LDPC) codes under iterative decoding over erasure channels. We derive several results on the asymptotic behavior of stopping sets in Tanner-graph ensembles, including the following. An expression for the normalized average stopping set distribution, yielding, in particular, a critical fraction of the block length above which codes have exponentially many stopping sets of that size. A relation between the degree distribution and the likely size of the smallest nonempty stopping set, showing that for a /spl radic/1-/spl lambda/'(0)/spl rho/'(1) fraction of codes with /spl lambda/'(0)/spl rho/'(1)2, the smallest nonempty stopping set is linear in the block length. Bounds on the average block error probability as a function of the erasure probability /spl epsi/, showing in particular that for codes with lowest variable degree 2, if /spl epsi/ is below a certain threshold, the asymptotic average block error probability is 1-/spl radic/1-/spl lambda/'(0)/spl rho/'(1)/spl epsi/.
The exact values of finite length average weight distributions for both binary ensembles and binary images of nonbinary ensembles of regular LDPC codes are computed. The exact average stopping set size distribution for the binary ensemble is also obtained. The computed spectra are applied in order to bound from above the average stopping redundancy of the ensemble of binary regular LDPC codes. The asymptotic typical normalized minimum distances for the binary image of the ensemble of nonbinary regular LDPC codes and the typical minimum stopping distances for the binary regular LDPC codes are also presented.
In this correspondence, we estimate the variance of weight and stopping set distribution of regular low-density parity-check (LDPC) ensembles. Using this estimate and the second moment method we obtain bounds on the probability that a randomly chosen code from regular LDPC ensemble has its weight distribution and stopping set distribution close to respective ensemble averages. We are able to show that a large fraction of total number of codes have their weight and stopping set distribution close to the average
This work presents an ensemble of irregular low-density parity check (LDPC) codes based on an ensemble of bipartite graphs, which formulated the average weight distribution. An analysis of the error performance of LDPC codes over a binary erasure channel together with an iterative decoding algorithm based on belief propagation has been clarified that a notion of stopping sets and their distribution with the asymptotic expression are derived. An irregular LDPC code ensemble that exhibits better performance in the sense of threshold is obtained by density evolution and explicitly formulates weight and stopping set distributions are defined.