On decoding iterated codes

Communications in Computer and Information Science, 2011

 Abstract-We show that for (systematic) linear codes the time complexity of unique decoding is     2/ 2 nRH R O n q  and the time complexity of minimum distance decoding is     2 nRH R O n q  . The proposed algorithm inspects all error patterns in the information set of the received message of weight less than 2 d or d , respectively. Index Terms-nearest neighbor decoding, unique decoding, bounded distance decoding, minimum distance decoding.

arXiv (Cornell University), 2020

Interleaved Reed-Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed-Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either fail to return a codeword or miscorrect to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed-Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.'s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known.

IEEE Transactions on Information Theory, 1985

ABSTRACT Complete decoding of systematic linear block codes by standard arrays with coset leaders not necessarily of minimal weight gives, in several interesting cases on the binary symmetric channel with error probability p , a smaller symbol error probability than decoding using the coset leaders of minimum weight. The proposed choice of the coset leaders allows for the classes of uniformly packed and nearly perfect binary codes to achieve a lower information bit error probability, at least for p sufficiently near zero.

2003

Iterative decoding of block codes is a rather old subject that regained much interest recently. The main idea behind iterative decoding is to break up the decoding problem into a sequence of stages, iterations, such that each stage utilizes the output from the previous stages to formulate its own result. In order for the iterative decoding algorithms to be practically feasible, the complexity in each stage, in terms of number of operations and hardware complexity, should be much less than that for the original non-iterative decoding problem. At the same time, the performance should approach the optimum, maximum likelihood decoding performance in terms of bit error rate.

The IMA Volumes in Mathematics and its Applications, 2001

Several popular, suboptimal algorithms for bit decoding of binary block codes such as turbo decoding, threshold decoding, and message passing for LDPC, were developed almost as a common sense approach to decoding of some specially designed codes. After their introduction, these algorithms have been studied by mathematical tools pertinent more to computer science than the conventional algebraic coding theory. We give an algebraic description of the optimal and suboptimal bit decoders and of the optimal and suboptimal message passing. We explain exactly how suboptimal algorithms approximate the optimal, and show how good this approximations are in some special cases.

IEEE Communications Letters, 2004

The sum-product iterative decoder, conventionally used for low-density parity-check (LDPC) codes, hold promise as a decoder for general linear block code decoding. However, the promise is only partly fulfilled because, as we show experimentally, the decoder performance degrades rapidly as a function of parity check matrix weight. Even in the case of decoder failure, however, we demonstrate that there is information present in the decoder output probabilities that can still help with the decoding problem.

IEEE Transactions on Information Theory, 1994

To decode a long block code with a large minimum distance by maximum likelihood decoding is practically impossible because the decoding complexity is simply enormous. However, if a code can be decomposed into constituent codes with smaller dimensions and simpler structure, it is possible to devise a practical and yet efficient scheme to decode the code. This paper investigates a class of decomposable codes, their distance and structural properties. It is shown that this class includes several classes of well-known and efficient codes as subclasses. Several methods for constructing decomposable codes or decomposing codes are presented. A two-stage (soft-decision or hard-decision) decoding scheme for decomposable codes, their translates or unions of translates is devised, and its error performance is analyzed for an AWGN channel. The two-stage soft-decision decoding is suboptimum. Error performances of some specific decomposable codes based on the proposed twostage soft-decision decoding are evaluated. It is shown that the proposed two-stage suboptimum decoding scheme provides an excellent trade-off between the error performance and decoding complexity for codes of moderate and long block length.

Journal of Algebra and Its Applications, 2013

We propose a decoding algorithm for the (u | u + v)-construction that decodes up to half of the minimum distance of the linear code. We extend this algorithm for a class of matrix-product codes in two different ways. In some cases, one can decode beyond the error-correction capability of the code.