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IEEE International Symposium on Information Theory (ISIT)

Sparse factor graph representations of reed-solomon and related codes

Jonathan S. Yedidia

Published Dec 23, 2004• 8 citations
Publisher

Abstract

We present sparse factor graph representations of Reed-Solomon codes based on a fast Fourier transform. These representations can be used to create encoders, or message-passing decoders that use soft input information. We discuss various simplifications and transformations of the factor graph representations that may be useful. Finally, we show that other interesting codes can be represented using sparse fast transform factor graphs.

DOI
10.1109/isit.2004.1365296

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We present sparse factor graph representations of Reed-Solomon codes based on a fast Fourier transform. These representations can be used to create encoders, or message-passing decoders that use soft input information. We discuss various simplifications and transformations of the factor graph representations that may be useful. Finally, we show that other interesting codes can be represented using sparse fast transform factor graphs.
We present a decoding algorithm for concatenated codes when the outer code is a Reed-Solomon code and the inner code is arbitrary. "Soft" information on the reliability of various symbols is passed by the inner decodings and exploited in the Reed-Solomon decoding. This is the first analysis of such a soft algorithm that works for arbitrary inner codes; prior analyses could only, handle some special inner codes. Crucial to our analysis is a combinatorial result on the coset weight distribution of codes given only its minimum distance. Our result enables us to decode essentially up to the "Johnson radius" of a concatenated code when the outer distance is large (the Johnson radius is the "a priori list decoding radius" of a code as a function of its distance). As a consequence, we are able to present simple and efficient constructions of q-ary linear codes that are list decodable up to a fraction (1 - 1/q - /spl epsiv/) of errors and have rate /spl Omega/(/spl epsiv//sup 6/). Codes that can correct such a large fraction of errors have found numerous complexity-theoretic applications. The previous constructions of linear codes with a similar rate used algebraic-geometric codes and thus suffered from a complicated construction and slow decoding.
. We introduce a new class of quantum error--correcting codes derived from (classical) Reed--Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform over finite fields are presented. 1 Introduction During the last years it has been shown that computers taking advantage of quantum mechanical phenomena outperform currently used computers. The striking examples are integer factoring in polynomial time (see [18]) and finding pre--images of an n--ary Boolean function ("searching") in time O( p 2 n ) (see [12]). Quantum computers are not only of theoretical nature---there are several suggestions how to physically realize them (see, e. g., [6, 7]). On the way towards building a quantum computer, one very important problem is to stabilize quantum mechanical systems since they are very vulnerable. A theory of quantum error--correcting codes has already been established (see [15]). Nevertheless, the pr...
This paper presents a soft decoding scheme based on the binary representations transferred from the parity-check matrices (PCMs) for Reed-Solomon (RS) codes. Referring to the modified binary PCM that has a systematic part and a high-density part corresponding to the least reliable variable nodes (LRVNs) and the most reliable variable nodes (MRVNs), respectively, an informed dynamic scheduling method, called Nested-Polling Residual Belief Propagation (NP-RBP), is applied to the corresponding Tanner graph. As with the popular adaptive BP (ABP) decoding approach, adaptation in a binary PCM based on the reliability of variable nodes is also conducted in the proposed NP-RBP decoding. The NP-RBP enables the LRVNs to receive significant updates and limits the correlation accumulation from the short cycles in the MRVNs. In order to enhance the error-rate performance for long codes, a bit-flipping (BF) technique is conducted in order to correct a selection of the errors in the MRVNs such that the propagation of these errors in the subsequent NP-RBP process can be avoided. The resultant decoder is termed NP-RBP-BF. For short codes such as the (31, 25) and (63, 55) RS codes, NP-RBP is able to provide an error-rate performance close to the maximum-likelihood (ML) bound. A more significant improvement can be observed for long codes. For instance, when the proposed NP-RBP-BF decoding is applied to the (255, 239) RS code, it can provide a gain of about 0.4 dB compared to the ABP decoding and the performance gap to the ML bound can be narrowed to about 0.25 dB at a frame error rate of2×10−32\times 10^{-3}2×10−3.
We introduce the concept of convolutional factor graphs, which represent convolutional factorizations of multivariate functions, just as conventional (multiplicative) factor graphs represent multiplicative factorizations. Convolutional and multiplicative factor graphs arise as natural Fourier transform duals. In coding theory applications, algebraic duality of group codes is essentially an instance of Fourier transform duality. Convolutional factor graphs arise when a code is represented as a sum of subcodes, just as conventional multiplicative factor graphs arise when a code is represented as an intersection of supercodes. With auxiliary variables, convolutional factor graphs give rise to "syndrome realizations" of codes, just as multiplicative factor graphs with auxiliary variables give rise to "state realizations." We introduce normal and co-normal extensions of a multivariate function, which essentially allow a given function to be represented with either a multiplicative or a convolutional factorization, as is convenient. We use these function extensions to derive a number of duality relationships among the corresponding factor graphs, and use these relationships to obtain the duality properties of Forney graphs as a special case.
Polar codes, introduced by Arıkan, are linear block codes that can achieve the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Polar codes of block length N are constructed by channel polarization method, which consists of channel combining and splitting operations to obtain N polarized subchannels from N copies of binary-input discrete memoryless channels. As N grows, symmetric channel capacities of the polarized subchannels converge to either 0 or 1. Polar codes are also close cousins of Reed-Muller codes and start to differ from each other for N≥32. Encoding and decoding of polar or Reed-Muller codes can be performed by using a factor graph, obtained from the n-th Kronecker product G=F^(⊗n) of F=[(1&0@1&1)] with N=2^n. Such a factor graph contains n=log⁡N stages; hence, by changing the order of stages with respect to each other, n! different factor graphs can be obtained. In the literature, some decoders using multiple factor graphs instead of a single factor graph are suggested. Therefore, it is of interest whether i) the K×N generator matrix of the code chosen by K active bits at the input of the encoder, and ii) the sum of the capacities of the K active channels that connect each input bit to the output vector of the encoder are invariant under stage permutations. In this study, we give an alternative proof of the fact that the answer to the first question is positive. It is also shown that the sum of the capacities of the K active channels is not invariant under stage permutations. Belief Propagation decoding performances on single and multiple factor graph decoders of polar and Reed-Muller codes over binary erasure channels are evaluated and compared. For multiple factor graph decoders, practical choice of factor graph sets that gives the best performance with low complexity is examined.