Symbolic Stochastic Chase Decoding of Reed-Solomon and BCH Codes

This paper proposes the Symbolic-Stochastic Chase Decoding Algorithm (S-SCA) for the Reed-Solomon (RS) and BCH codes. By efficient usage of void space between constellation points for $q$-ary modulations and using soft information at the input of the decoder, the S-SCA is capable of outperforming conventional Symbolic-Chase algorithm (S-CA) with less computational cost. Since the S-SCA starts with the randomized generation of likely test-vectors, it reduces the complexity to polynomial order and also it does not need to find the least reliable symbols to generate test-vectors. Our simulation results show that by increasing the number of test-vectors, the performance of the algorithm can approach the ML bound. The S-SCA($1K$) provides near $2$ dB gain in comparison with S-CA($1K$) for $(31, 25)$ RS code using $32$-QAM. Furthermore, the algorithm provides near $3$ dB further gain with $1K$ iteration compared with S-CA($65K$) when $(255, 239)$ RS code is used in an AWGN channel. For the Rayleigh fading channel and the same code, the algorithm provides more that $5$ dB gain. Also for $(63, 57)$ BCH codes and $8$-PSK modulation the proposed algorithm provides $3$dB gain with less complexity. This decoder is Soft-Input Soft-Output (SISO) decoder and is highly attractive in low power applications. Finally, the Symbolic-Search Bitwise-Transmission Stochastic Chase Algorithm (SSBT-SCA) was introduced for RS codes over BPSK transmission that is capable of generating symbolic test-vectors that reduce complexity and mitigate burst errors

New Set of Codes for the Maximum-Likelihood Decoding Problem

The maximum-likelihood decoding problem is known to be NP-hard for general linear and Reed-Solomon codes. In this paper, we introduce the notion of A-covered codes, that is, codes that can be decoded through a polynomial time algorithm A whose decoding bound is beyond the covering radius. For these codes, we show that the maximum-likelihood decoding problem is reachable in polynomial time in the code parameters. Focusing on bi- nary BCH codes, we were able to find several examples of A-covered codes, including two codes for which the maximum-likelihood decoding problem can be solved in quasi-quadratic time.Comment: in Yet Another Conference on Cryptography, Porquerolle : France (2010

Guessing random additive noise decoding with symbol reliability information (SRGRAND)

In computer communications, discrete data are first channel coded and then modulated into continuous signals for transmission and reception. In a hard detection setting, only demodulated data are provided to the decoder. If soft information on received signal quality is provided, its use can improve decoding accuracy. Incorporating it, however, typically comes at the expense of increased algorithmic complexity. Here we introduce a mechanism to use binarized soft information in the Guessing Random Additive Noise Decoding framework such that decoding accuracy is increased, but computational complexity is decreased. The principle envisages a code-book-independent quantization of soft information where demodulated symbols are additionally indicated to be reliable or unreliable. We introduce two algorithms that incorporate this information, one of which identifies a Maximum Likelihood (ML) decoding and the other either reports an ML decoding or an error. Both are suitable for use with any block-code, and are capacity-achieving. We determine error exponents and asymptotic complexity. They achieve higher rates with lower error probabilities and less algorithmic complexity than their hard detection counterparts. As practical illustrations, we compare performance with majority logic decoding of a Reed-Muller code, with Berlekamp-Massey decoding of a BCH code, and establish performance of Random Linear Codes

Algebraic Decoding for Doubly Cyclic Convolutional Codes

An iterative decoding algorithm for convolutional codes is presented. It successively processes $N$ consecutive blocks of the received word in order to decode the first block. A bound is presented showing which error configurations can be corrected. The algorithm can be efficiently used on a particular class of convolutional codes, known as doubly cyclic convolutional codes. Due to their highly algebraic structure those codes are well suited for the algorithm and the main step of the procedure can be carried out using Reed-Solomon decoding. Examples illustrate the decoding and a comparison with existing algorithms is being made.Comment: 16 page

Code Constructions based on Reed-Solomon Codes

Reed--Solomon codes are a well--studied code class which fulfill the Singleton bound with equality. However, their length is limited to the size $q$ of the underlying field $\mathbb{F}_q$. In this paper we present a code construction which yields codes with lengths of factors of the field size. Furthermore a decoding algorithm beyond half the minimum distance is given and analyzed

Algebraic Soft Decoding of Reed-Solomon Codes Using Module Minimization

The interpolation based algebraic decoding for Reed-Solomon (RS) codes can correct errors beyond half of the code's minimum Hamming distance. Using soft information, the algebraic soft decoding (ASD) further improves the decoding performance. This paper presents a unified study of two classical ASD algorithms in which the computationally expensive interpolation is solved by the module minimization (MM) technique. An explicit module basis construction for the two ASD algorithms will be introduced. Compared with Koetter's interpolation, the MM interpolation enables the algebraic Chase decoding and the Koetter-Vardy decoding perform less finite field arithmetic operations. Re-encoding transform is applied to further reduce the decoding complexity. Computational cost of the two ASD algorithms as well as their re-encoding transformed variants are analyzed. This research shows re-encoding transform attributes to a lower decoding complexity by reducing the degree of module generators. Furthermore, Monte-Carlo simulation of the two ASD algorithms has been performed to show their decoding and complexity competency.Comment: 30 pages, 4 figure

An Algebraic Framework for Concatenated Linear Block Codes in Side Information Based Problems

This work provides an algebraic framework for source coding with decoder side information and its dual problem, channel coding with encoder side information, showing that nested concatenated codes can achieve the corresponding rate-distortion and capacity-noise bounds. We show that code concatenation preserves the nested properties of codes and that only one of the concatenated codes needs to be nested, which opens up a wide range of possible new code combinations for these side information based problems. In particular, the practically important binary version of these problems can be addressed by concatenating binary inner and non-binary outer linear codes. By observing that list decoding with folded Reed- Solomon codes is asymptotically optimal for encoding IID q-ary sources and that in concatenation with inner binary codes it can asymptotically achieve the rate-distortion bound for a Bernoulli symmetric source, we illustrate our findings with a new algebraic construction which comprises concatenated nested cyclic codes and binary linear block codes

Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined. Index Terms Bound on the minimum distance, burst error, efficient decoding, folded code, repeated-root cyclic code, repeated-root cyclic product cod

Prefactor Reduction of the Guruswami-Sudan Interpolation Step

The concept of prefactors is considered in order to decrease the complexity of the Guruswami-Sudan interpolation step for generalized Reed-Solomon codes. It is shown that the well-known re-encoding projection due to Koetter et al. leads to one type of such prefactors. The new type of Sierpinski prefactors is introduced. The latter are based on the fact that many binomial coefficients in the Hasse derivative associated with the Guruswami-Sudan interpolation step are zero modulo the base field characteristic. It is shown that both types of prefactors can be combined and how arbitrary prefactors can be used to derive a reduced Guruswami-Sudan interpolation step.Comment: 13 pages, 3 figure

Bounds on the ML Decoding Error Probability of RS-Coded Modulation over AWGN Channels

This paper is concerned with bounds on the maximum-likelihood (ML) decoding error probability of Reed-Solomon (RS) codes over additive white Gaussian noise (AWGN) channels. To resolve the difficulty caused by the dependence of the Euclidean distance spectrum on the way of signal mapping, we propose to use random mapping, resulting in an ensemble of RS-coded modulation (RS-CM) systems. For this ensemble of RS-CM systems, analytic bounds are derived, which can be evaluated from the known (symbol-level) Hamming distance spectrum. Also presented in this paper are simulation-based bounds, which are applicable to any specific RS-CM system and can be evaluated by the aid of a list decoding (in the Euclidean space) algorithm. The simulation-based bounds do not need distance spectrum and are numerically tight for short RS codes in the regime where the word error rate (WER) is not too low. Numerical comparison results are relevant in at least three aspects. First, in the short code length regime, RS-CM using BPSK modulation with random mapping has a better performance than binary random linear codes. Second, RS-CM with random mapping (time varying) can have a better performance than with specific mapping. Third, numerical results show that the recently proposed Chase-type decoding algorithm is essentially the ML decoding algorithm for short RS codes.Comment: arXiv admin note: text overlap with arXiv:1309.1555 by other author