137 research outputs found
A simple algorithm for decoding Reed-Solomon codes and its relation to the Welch-Berlekamp algorithm
A simple and natural Gao algorithm for decoding algebraic codes is described.
Its relation to the Welch-Berlekamp and Euclidean algorithms is given.Comment: 7 pages. Submitted to IEEE Transactions on Information Theor
Re-encoding reformulation and application to Welch-Berlekamp algorithm
The main decoding algorithms for Reed-Solomon codes are based on a bivariate
interpolation step, which is expensive in time complexity. Lot of interpolation
methods were proposed in order to decrease the complexity of this procedure,
but they stay still expensive. Then Koetter, Ma and Vardy proposed in 2010 a
technique, called re-encoding, which allows to reduce the practical running
time. However, this trick is only devoted for the Koetter interpolation
algorithm. We propose a reformulation of the re-encoding for any interpolation
methods. The assumption for this reformulation permits only to apply it to the
Welch-Berlekamp algorithm
Reed-Solomon list decoding from a system-theoretic perspective
In this paper, the Sudan-Guruswami approach to list decoding of Reed-Solomon (RS) codes is cast in a system-theoretic framework. With the data, a set of trajectories or time series is associated which is then modeled as a so-called behavior. In this way, a connection is made with the behavioral approach to system theory. It is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the Sudan-Guruswami approach. The concept of "weighted row reduced" is introduced and used to achieve minimality. Two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given
Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes
For the majority of the applications of Reed-Solomon (RS) codes, hard
decision decoding is based on syndromes. Recently, there has been renewed
interest in decoding RS codes without using syndromes. In this paper, we
investigate the complexity of syndromeless decoding for RS codes, and compare
it to that of syndrome-based decoding. Aiming to provide guidelines to
practical applications, our complexity analysis differs in several aspects from
existing asymptotic complexity analysis, which is typically based on
multiplicative fast Fourier transform (FFT) techniques and is usually in big O
notation. First, we focus on RS codes over characteristic-2 fields, over which
some multiplicative FFT techniques are not applicable. Secondly, due to
moderate block lengths of RS codes in practice, our analysis is complete since
all terms in the complexities are accounted for. Finally, in addition to fast
implementation using additive FFT techniques, we also consider direct
implementation, which is still relevant for RS codes with moderate lengths.
Comparing the complexities of both syndromeless and syndrome-based decoding
algorithms based on direct and fast implementations, we show that syndromeless
decoding algorithms have higher complexities than syndrome-based ones for high
rate RS codes regardless of the implementation. Both errors-only and
errors-and-erasures decoding are considered in this paper. We also derive
tighter bounds on the complexities of fast polynomial multiplications based on
Cantor's approach and the fast extended Euclidean algorithm.Comment: 11 pages, submitted to EURASIP Journal on Wireless Communications and
Networkin
A comparison of VLSI architectures for time and transform domain decoding of Reed-Solomon codes
It is well known that the Euclidean algorithm or its equivalent, continued fractions, can be used to find the error locator polynomial needed to decode a Reed-Solomon (RS) code. It is shown that this algorithm can be used for both time and transform domain decoding by replacing its initial conditions with the Forney syndromes and the erasure locator polynomial. By this means both the errata locator polynomial and the errate evaluator polynomial can be obtained with the Euclidean algorithm. With these ideas, both time and transform domain Reed-Solomon decoders for correcting errors and erasures are simplified and compared. As a consequence, the architectures of Reed-Solomon decoders for correcting both errors and erasures can be made more modular, regular, simple, and naturally suitable for VLSI implementation
Some Applications of Coding Theory in Computational Complexity
Error-correcting codes and related combinatorial constructs play an important
role in several recent (and old) results in computational complexity theory. In
this paper we survey results on locally-testable and locally-decodable
error-correcting codes, and their applications to complexity theory and to
cryptography.
Locally decodable codes are error-correcting codes with sub-linear time
error-correcting algorithms. They are related to private information retrieval
(a type of cryptographic protocol), and they are used in average-case
complexity and to construct ``hard-core predicates'' for one-way permutations.
Locally testable codes are error-correcting codes with sub-linear time
error-detection algorithms, and they are the combinatorial core of
probabilistically checkable proofs
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