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IEEE Transactions on Communications

Performance of Product Codes and Related Structures with Iterated Decoding

JØrn Justesen

Published Dec 22, 2010• 91 citations
Publisher

Abstract

Several modifications of product codes have been suggested as standards for optical networks. We show that the performance exhibits a threshold that can be estimated from a result about random graphs. For moderate input bit error probabilities, the output error rates for codes of finite length can be found by easy simulations. The analysis indicates that the performance curve can be extrapolated until the error floor is reached. The analysis allows us to calculate the error floors and avoid time-consuming simulations.

DOI
10.1109/tcomm.2011.121410.090146

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Several modifications of product codes have been suggested as standards for optical networks. We show that the performance exhibits a threshold that can be estimated from a result about random graphs. For moderate input bit error probabilities, the output error rates for codes of finite length can be found by easy simulations. The analysis indicates that the performance curve can be extrapolated until the error floor is reached. The analysis allows us to calculate the error floors and avoid time-consuming simulations.
Several modifications of product codes have been suggested as standards for optical networks. We show that the performance exhibits a threshold that can be estimated from a result about random graphs. For moderate input bit error probabilities, the output error rates for codes of finite length can be found by easy simulations. The analysis indicates that the performance curve can be extrapolated until the error floor is reached. The analysis allows us to calculate the error floors and avoid time-consuming simulations. Index Terms—Product codes, iterative decoding, optical net- works.
Products of Reed-Solomon codes are important in applications because they offer a combination of large blocks, low decoding complexity, and good performance. A recent result on random graphs can be used to show that with high probability a large number of errors can be corrected by iterating minimum distance decoding. We present an analysis related to density evolution which gives the exact asymptotic value of the decoding threshold and also provides a closed form approximation to the distribution of errors in each step of the decoding of finite length codes.
Performance analysis for iterative decoders: In Chapter 7, we discuss the behavior of LDPC codes under iterative erasure decoding. When the blocklength goes to infinity, for many LDPC codes, the symbol error probability exhibits a so-called threshold phenomenon that is, there exists a certain channel erasure probability below which error-free communication is possible, while this is not guaranteed above it. We discuss how to compute this threshold for ensembles of LDPC codes on memoryless erasure channels. In the finite-length setting, one may observe a flattening of the symbol error rate curve owing to stopping sets – specific structures in the code’s bipartite graph. Knowing their number and size allows predicting this so-called error floor. Based on our findings, we discuss how to design good LDPC for memoryless erasure channels with extension to channels with memory.
We present a bit error rate analysis for iterative two-stage hard decision decoding of product codes and proposed a concatenated codes suitable for optical communication. Our method is based on an importance sampling technique characterized by recursive calculation of cumulative binomial probabilities and direct biasing of error patterns. We can accelerate the Monte Carlo simulations by several orders of magnitude such as 10/sup 8/ and estimate the coding gains at the BERs as low as 10/sup -15/ required in optical communications. We apply the method for comparative BER analysis of product codes and proposed concatenated codes and show that product codes of small minimum distance have error floors at low BERs while the proposed concatenated codes of large minimum distance do not.
The bit-error-rate (BER) performance of 3-dimensional (3D) product codes under iterative bounded-distance decoding of the component codes is considered and a framework for analyzing the BER-performance is presented. The performance analysis of iterative decoding is based on a graphical model of the underlying 3D product code, which is a tripartite 3-uniform hypergraph. The BER performance for 3D product codes shows a threshold behavior. The thresholds can be approximated by an exit-chart-like technique from the parameters of the component codes. In the case of a symmetric product code, for which all three component codes are based on the same lineart-error correcting code, asymptotically for large code lengths, the BER-threshold is determined by the threshold for the appearance of ak-core withk=t+ 1 in the graphical model.
Low-density parity-check (LDPC) codes have the potential for applications in future high throughput optical communications due to their significant error correction capability and the parallel decoding. However, they are not able to satisfy the very low bit error rate (BER) requirement due to the error floor phenomenon. In this paper, we propose a low-complexity iterative decoding scheme for product codes consisting of very high rate outer codes and LDPC codes. The outer codes aim at eliminating the residual error floor of LDPC codes with quite low implementation costs. Furthermore, considering the long simulation time of computer simulation for evaluating very low BER, the hardware platform is built to accelerate the evaluation of the proposed iterative decoding methods. Simultaneously, the fixed-point effects of the decoding algorithms are also be evaluated. The experimental results show that the iterative decoding of the product codes can achieve a quite low bit error rate. The evaluation using field programmable gate array (FPGA) also proves that product codes with LDPC codes and high-rate algebraic codes can achieve a good trade-off between complexity and throughput.