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.
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.
We consider an ensemble of random q -ary LDPC codes. As constituent codes, we use q -ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.
We consider hard-decision iterative decoders for product codes over the erasure channel, which employ repeated rounds of decoding rows and columns alternatingly. We derive the exact asymptotic probability of decoding failure as a function of the error-correction capabilities of the row and column codes, the number of decoding rounds, and the channel erasure probability. We examine both the case of codes capable of correcting a constant amount of errors, and the case of codes capable of correcting a constant fraction of their length
Reed-Muller (RM) codes encode an m-variate polynomial of degree at most r by evaluating it on all points in {0,1}m. We denote this code by RM(r,m). The minimum distance of RM(r,m) is 2m-rand so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length n=2m) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree r=o(√m)), we can correct a random set of (1/2-o(1))n errors with high probability. For high rate codes (of degree m-r for r=o(√m/log m)), we can correct roughly mr/2errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m-(2r+2),m), for which the same erasure pattern can be corrected in RM(m-(r+1),m). The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kötter that generalizes the Berlekamp-Welch algorithm for decoding Reed-Solomon codes.
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.
Reed-Muller codes encode an m-variate polynomial of degree r by evaluating it on all points in {0,1}m. We denote this code by RM(m,r). The minimal distance of RM(m,r) is 2m−r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this work we give an efficient algorithm (in the block length n=2m) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree r=o(m)) we can correct a random set of (1/2−o(1))n errors with high probability. For high rate codes (of degree m−r for r=o(m/logm)), we can correct roughly mr/2 errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m,m−(2r+2)) for which the same erasure pattern can be corrected in RM(m,m−(r+1)). The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon codes.
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.