Showing papers on "List decoding published in 1972"
TL;DR: It is shown that as the signal-to-noise ratio (SNR) increases, the asymptotic behavior of these decoding algorithms cannot be improved, and computer simulations indicate that even for SNR the performance of a correlation decoder can be approached by relatively simple decoding procedures.
Abstract: A class of decoding algorithms that utilizes channel measurement information, in addition to the conventional use of the algebraic properties of the code, is presented. The maximum number of errors that can, with high probability, be corrected is equal to one less than d , the minimum Hamming distance of the code. This two-fold increase over the error-correcting capability of a conventional binary decoder is achieved by using channel measurement (soft-decision) information to provide a measure of the relative reliability of each of the received binary digits. An upper bound on these decoding algorithms is derived, which is proportional to the probability of an error for d th order diversity, an expression that has been evaluated for a wide range of communication channels and modulation techniques. With the aid of a lower bound on these algorithms, which is also a lower bound on a correlation (maximum-likelihood) decoder, we show for both the Gaussian and Rayleigh fading channels, that as the signal-to-noise ratio (SNR) increases, the asymptotic behavior of these decoding algorithms cannot be improved. Computer simulations indicate that even for !ow SNR the performance of a correlation decoder can be approached by relatively simple decoding procedures. In addition, we study the effect on the performance of these decoding algorithms when a threshold is used to simplify the decoding process.
1,165 citations
TL;DR: The analysis shows further that the "natural" choice of bias in the metric is the code rate and gives insight into why the Fano metric has proved to be the best practical choice in sequential decoding.
Abstract: It is shown that the metric proposed originally by Fano for sequential decoding is precisely the required statistic for minimum-error-probability decoding of variable-length codes. The analysis shows further that the "natural" choice of bias in the metric is the code rate and gives insight into why the Fano metric has proved to be the best practical choice in sequential decoding. The recently devised Jelinek-Zigangirov "stack algorithm" is shown to be a natural consequence of this interpretation of the Fano metric. Finally, it is shown that the elimination of the bias in the "truncated" portion of the code tree gives a slight reduction in average computation at the sacrifice of increased error probability.
134 citations
TL;DR: A decoding algorithm is given that can correct up to the number of errors guaranteed by the product minimum distance, rather than about half that number when the iterated codes are decoded independently.
Abstract: We give a decoding algorithm for iterated codes that can correct up to the number of errors guaranteed by the product minimum distance, rather than about half that number when the iterated codes are decoded independently. This result is achieved by adapting Forney's generalized minimum distance decoding for use with iterated codes. We derive results on the simultaneous burst- and random-error-correction capability of iterated codes that improve considerably on known results.
69 citations
TL;DR: In this correspondence, a decoding algorithm to decode beyond the BCH bound is introduced and gives a complete minimum distance decoding for any cyclic code.
Abstract: In this correspondence, a decoding algorithm to decode beyond the BCH bound is introduced. It gives a complete minimum distance decoding for any cyclic code. A comparison between this decoding algorithm and previously existing ones is also given.
44 citations
TL;DR: In this article, the burst-b distance between two binary vectors is defined and shown to be a metric for binary-input, Q-ary output channels where errors occur in bursts.
Abstract: The burst- b distance between two binary vectors is defined and shown to be a metric. This definition is applied to a binary-input, Q -ary output channel where errors occur in bursts. A decoding algorithm is presented for such a channel that is an extension of Weldon's [2] weighted erasure decoding. Examples are presented illustrating the techniques.
21 citations
TL;DR: In this correspondence some classes of unequal protection codes are constructed utilizing difference sets or triangles using threshold decoding, and these codes use threshold decoding.
Abstract: In this correspondence some classes of unequal protection codes are constructed utilizing difference sets or triangles. These codes use threshold decoding.
20 citations
TL;DR: Two error-erasure decoding algorithms for product codes that correct all the error- erasure patterns guaranteed correctable by the minimum Hamming distance of the product code are given.
Abstract: Two error-erasure decoding algorithms for product codes that correct all the error-erasure patterns guaranteed correctable by the minimum Hamming distance of the product code are given. The first algorithm works when at least one of the component codes is majority-logic decodable. The second algorithm works for any product code. Both algorithms use the decoders of the component codes.
15 citations
TL;DR: This correspondence presents a decoding procedure for finite geometry codes that requires as few decoding steps as possible and it is shown that the minimum number of steps is a logarithmic function of the dimension of the associated geometry.
Abstract: In a recent paper [1], techniques for reducing the number of majority-logic decoding steps for finite geometry codes have been proposed. However, the lower bound of [1, lemma 4] is incorrect; finite geometry codes, in general, cannot be decoded in less than or equal to three steps of orthogonalization, as was claimed. This correspondence presents a decoding procedure for finite geometry codes that requires as few decoding steps as possible. It is shown that the minimum number of steps is a logarithmic function of the dimension of the associated geometry.
15 citations
Journal Article•
TL;DR: The burst-b distance between two binary vectors is defined and shown to be a metric and a decoding algorithm is presented for such a channel that is an extension of Weldon's weighted erasure decoding.
Abstract: The burst-b distance between two binary vectors is defined and shown to be a metric. This definition is applied to a binary-input, Q-ary output channel where errors occur in bursts. A decoding algorithm is presented for such a channel that is an extension of Weldon's (1971) weighted erasure decoding. Examples are presented illustrating the techniques.
15 citations
TL;DR: A new form of linear programming is presented that minimizes the real sum of zero-one variables under constraints that are linear in the field of binary numbers and thus nonlinear in theField of real numbers.
12 citations
TL;DR: Upper bounds are derived on the error probability that can be achieved by using the maximum-likelihood algorithm of sequential decoding for the binary symmetric channel using constraint length and backsearch limit.
Abstract: Upper bounds are derived on the error probability that can be achieved by using the maximum-likelihood algorithm of sequential decoding for the binary symmetric channel. The bounds are functions of constraint length and backsearch limit.
Journal Article•
TL;DR: Two error-erasure decoding algorithms for product codes that correct all the error- erasure patterns guaranteed correctable by the minimum Hamming distance of the product code are given.
Abstract: Two error-erasure decoding algorithms for product codes that correct all the error-erasure patterns guaranteed correctable by the minimum Hamming distance of the product code are given. The first algorithm works when at least one of the component codes is majority-logic decodable. The second algorithm works for any product code. Both algorithms use the decoders of the component codes.
TL;DR: It is shown that any decoding function for a linear binary code can be realized as a weighted majority of nonorthogonal parity checks.
Abstract: It is shown that any decoding function for a linear binary code can be realized as a weighted majority of nonorthogonal parity checks An example is given of a four-error-correcting code that is neither L-step orthogonalizable nor one-step majority decodable using non-orthogonal parity checks and yet is one-step weighted-majority decodable using only ten nonorthogonal parity checks
TL;DR: The error-propagation problem that is encountered in decoding noncatastrophic codes with feedback decoders is analyzed and it is shown that if thc decoding depth is too short, unlimited error propagation will occur and that the depth can be chosen long enough so that unlimited error propagate does not occur.
Abstract: In this correspondence we analyze the error-propagation problem that is encountered in decoding noncatastrophic codes with feedback decoders. We show that if thc decoding depth is too short, unlimited error propagation will occur and that the depth can be chosen long enough so that unlimited error propagation does not occur when minimum-distance feedback decoders are used.
TL;DR: An algorithm is presented for the decoding of triple-error-correcting binary b.c.h that is particularly suitable for parallel implementation and requires no invertors.
Abstract: An algorithm is presented for the decoding of triple-error-correcting binary b.c.h, codes. The algorithm is particularly suitable for parallel implementation and requires no invertors.
TL;DR: A stochastic model is described for the decoder of an optimal burst-correcting convolutional code and an upper bound is obtained for \bar{p} , the error probability per word after decoding.
Abstract: A stochastic model is described for the decoder of an optimal burst-correcting convolutional code. From this model, an upper bound is obtained for \bar{p} , the error probability per word after decoding.
TL;DR: If one of the component codes is decodable using Rudolph's nonorthogonal algorithm, and the other using Reed-Massey's L -step orthogonalization, then the resulting product code is majority-logic decodability in L steps, where nonorthogsonal checks are used in the first step and orthogomatic checks in the remaining steps.
Abstract: In this correspondence additional results on majority-logic decoding of product codes are presented. If one of the component codes is decodable using Rudolph's nonorthogonal algorithm, and the other using Reed-Massey's L -step orthogonalization, then the resulting product code is majority-logic decodable in L steps, where nonorthogonal checks are used in the first step and orthogonal checks in the remaining steps. If the component codes are both decodable using Rudolph's algorithm, the product code is majority-logic decodable using the same procedure.
01 Jun 1972
TL;DR: A partitioned 3 state Gilbert model is used to model a burst channel and a method of calculating error sequence probabilities using this model is introduced, and Observations are made about the general type of decoding rule to use to give the lowest probability of decoding error on burst channels when using an interleaving technique.
Abstract: : A brief discussion of basic encoding and decoding on noisy channels is presented to provide a background for the experimental portion of this research. A partitioned 3 state Gilbert model is used to model a burst channel and a method of calculating error sequence probabilities using this model is introduced. Error sequence probability calculations are made using a (7.3) maximal length code and a (15,7) BCH code. Observations are made about the general type of decoding rule to use to give the lowest probability of decoding error on burst channels when using an interleaving technique.
01 Aug 1972
TL;DR: The total set of performance equations for error-detecting-and-correcting codes in a random-error environment is developed and establish probabilistic rates for the occurrence of decoding conditions such as nondetection, miscorrection (erroneous data reconstruction), decoding failure and correct decoding for a given code.
Abstract: A method of analyzing the performance of error-detecting-and-correcting codes in a random-error environment is presented. A tutorial section is included which establishes the general background required for the analysis. The subject of error detection and nondetection is covered extensively in existing literature. The concept of erroneous data reconstruction is almost totally avoided in code performance discussions. The total set of performance equations for error-detecting-and-correcting codes in a random-error environment is developed. These equations establish probabilistic rates for the occurrence of decoding conditions such as nondetection, miscorrection (erroneous data reconstruction), decoding failure and correct decoding for a given code. Examples illustrating the performance of several codes are given.