Showing papers on "List decoding published in 1971"
TL;DR: It is shown that a significant improvement in the performance with respect to other methods is achievable by the maximum likelihood decoding method and can reduce raw error ratehse in 10-3 to 10-4 range by a factor of 50 to 300.
Abstract: A digital magnetic recording system is viewed in this paper as a linear system that inherently includes a correlative level encoder. This encoder can be regarded aas linear finite-state machine like a convolutional encoder. The maximum likelihood decoding method recently devised by Viterbi to decode convolutional codes is then applietdo digital magnetic recording systems. The decoding algorithm and its implementation are discussed in detail.
Expressions for the decoding error probability are obtained and confirmed by computer simulations. It is shown that a significant improvement in the performance with respect to other methods is achievable by the maximum likelihood decoding method. For example, under the Gaussian noise assumption the proposed technique can reduce raw error ratehse in 10-3 to 10-4 range by a factor of 50 to 300. These results indicate that the maximum likelihood decoding method gains as much as 2.5 dB in signal-to-noise ratio over the conventional bit-by-bit detection method.
175 citations
IBM1
TL;DR: An application of the maximum-likelihood decoding (MLD) algorithm, which was originally proposed by Viterbi in decoding convolutional codes, is discussed and it is shown that a substantial performance gain is attainable by this probabilistic decoding method.
Abstract: Modems for digital communication often adopt the so-called correlative level coding or the partial-response signaling, which attains a desired spectral shaping by introducing controlled intersymbol interference terms. In this paper, a correlative level encoder is treated as a linear finite-state machine and an application of the maximum-likelihood decoding (MLD) algorithm, which was originally proposed by Viterbi in decoding convolutional codes, is discussed. Asymptotic expressions for the probability of decoding error are obtained for a class of correlative level coding systems, and the results are confirmed by computer simulations. It is shown that a substantial performance gain is attainable by this probabilistic decoding method.
141 citations
TL;DR: The iterative algorithm for decoding binary BCH codes presented by Berlekamp and, in an alternative form, by Massey is modified to eliminate inversion.
Abstract: The iterative algorithm for decoding binary BCH codes presented by Berlekamp and, in an alternative form, by Massey is modified to eliminate inversion. Because inversion in a finite field is time consuming and requires relatively complex circuitry, this new algorithm should he useful in practical applications of multiple-error-correcting binary BCH codes.
109 citations
TL;DR: A class of rate 1/2 nonsystematic convolutional codes with an undetected decoding error probability verified by simulation to be much smaller than for the best systematic codes of the same constraint length, and a "quick-look-in" feature that permits recovery of the information sequence from the hard-decisioned received data without decoding simply by modulo-two addition of the received sequences.
Abstract: Previous space applications of sequential decoding have all employed convolutional codes of the systematic type where the information sequence itself is used as one of the encoded sequences. This paper describes a class of rate 1/2 nonsystematic convolutional codes with the following desirable properties: 1) an undetected decoding error probability verified by simulation to be much smaller than for the best systematic codes of the same constraint length; 2) computation behavior with sequential decoding verified by simulation to be virtually identical to that of the best systematic codes; 3) a "quick-look-in" feature that permits recovery of the information sequence from the hard-decisioned received data without decoding simply by modulo-two addition of the received sequences; and 4) suitability for encoding by simple circuitry requiring less hardware than encoders for the best systematic codes of the same constraint length. Theoretical analyses are given to show 1) that with these codes the information sequence is extracted as reliably as possible without decoding for nonsystematic codes and 2) that the constraints imposed to achieve the quicklook-in feature do not significantly limit the error-correcting ability of the codes in the sense that the Gilbert bound on minimum distance can still be attained under these constraints. These codes have been adopted for use in several forthcoming space missions.
79 citations
TL;DR: A method of constructing a new class of majority-logic decodable block codes is presented, which has more information digits than the Reed-Muller codes of the same length and the same minimum distance.
Abstract: The attractiveness of majority-logic decoding is its simple implementation Several classes of majority-logic decodable block codes have been discovered for the past two decades In this paper, a method of constructing a new class of majority-logic decodable block codes is presented Each code in this class is formed by combining majority-logic decodable codes of shorter lengths A procedure for orthogonalizing codes of this class is formulated For each code, a lower bound on the number of correctable errors with majority-logic decoding is obtained An upper bound on the number of orthogonalization steps for decoding each code is derived Several majority-logic decodable codes that have more information digits than the Reed-Muller codes of the same length and the same minimum distance are found Some results presented in this paper are extensions of the results of Lin and Weldon [11] and Gore [12] on the majority-logic decoding of direct product codes
51 citations
TL;DR: It is shown how nonsystematic Reed-Solomon (RS) codes encoded by means of the Chinese remainder theorem can be decoded using the Berlekamp algorithm.
Abstract: It is shown how nonsystematic Reed-Solomon (RS) codes encoded by means of the Chinese remainder theorem can be decoded using the Berlekamp algorithm. The Chien search and calculation of error values are not needed but are replaced by a polynomial division and added calculation in determining the syndrome. It is shown that for certain cases of low-rate RS codes, the total decoding computation may be less than the usual method used with cyclic codes. Encoding and decoding for shorter length codes is presented.
50 citations
TL;DR: This paper presents an algebraic technique for decoding binary block codes in situations where the demodulator quantizes the received signal space into Q > 2 regions, applicable in principle to any block code for which a binary decoding procedure is known.
Abstract: This paper presents an algebraic technique for decoding binary block codes in situations where the demodulator quantizes the received signal space into Q > 2 regions. The method, referred to as weighted erasure decoding (WED), is applicable in principle to any block code for which a binary decoding procedure is known.
47 citations
IBM1
TL;DR: Analytical and simulation results on the performance of the proposed decoding scheme are presented and an asymptotic expression for the decoding error rate is derived in closed form as a function of the channel signal-to-noise ratio.
Abstract: Decoding of a correlative level coding or partialresponse signaling system is discussed in an algebraic framework. A correction scheme in which the quantizer Output includes ambiguity levels is proposed. The implementation and algorithm of error correction is discussed in some detail. An optimum design of the quantizer based on Chow's earlier work is discussed. Both analytical and simulation results on the performance of the proposed decoding scheme are presented. An asymptotic expression for the decoding error rate is derived in closed form as a function of the channel signal-to-noise ratio. This is also compared with the conventional bit-by-bit detection method and the maximumlikelihood decoding method recently studied.
36 citations
TL;DR: A new method of decoding is presented that utilizes algebraic constraints across streams of convolutionally encoded information sequences to improve the performance of ordinary sequential decoding and over the older hybrid scheme developed by Falconer.
Abstract: Hybrid decoding technique for symmetrical binary input channels, using bootstrap algorithm across convolutionally encoded information streams
28 citations
TL;DR: The computational work and the time required to decode with reliability E at code rate R on noisy channels are defined, and bounds on the size of these measures are developed.
Abstract: The computational work and the time required to decode with reliability E at code rate R on noisy channels are defined, and bounds on the size of these measures are developed. A number of ad hoc decoding procedures are ranked on the basis of the computational work they require.
TL;DR: In this paper, the authors introduced generalized shifted linear codes (SLC) which can achieve a positive rate on arbitrary discrete memoryless channels if Shannon's capacity is positive and the length of the alphabet is less or equal to 5.
Abstract: Summary It was proved by Ahlswede (1971) that codes whose codewords form a group or even a linear space do not achieve Shannon's capacity for discrete memoryless channels even if the decoding procedure is arbitrary Sharper results were obtained in Part I of this paper For practical purposes, one is interested not only in codes which allow a short encoding procedure but also an efficient decoding procedure Linear codes—the codewords form a linear space and the decoding is done by coset leader decoding — have a fairly efficient decoding procedure But in order to achieve high rates the following slight generalization turns out to be very useful: We allow the encoder to use a coset of a linear space as a set of codewords We call these codes shifted linear codes or coset codes They were implicitly used by Dobrushin (1963) This new code concept has all the advantages of the previous one with respect to encoding and decoding efficiency and enables us to achieve positive rate on discrete memoryless channels whenever Shannon's channel capacity is positive and the length of the alphabet is less or equal to 5 (Theorem 311) (The result holds very likely also for all alphabets with a length a = ps, p prime, s positive integer) A disadvantage of the concepts of linear codes and of shifted linear codes is that they can be defined only for alphabets whose length is a prime power In order to overcome this difficulty, we introduce generalized shifted linear codes With these codes we can achieve a positive rate on arbitrary discrete memoryless channels if Shannon's capacity is positive (Theorem 321)
TL;DR: An improved decoding algorithm for codes that are constructed from finite geometries is introduced and it is shown that these codes can be orthogonalized in less than or equal to three steps, ensuring majority-logic decodable codes.
Abstract: In this paper, an improved decoding algorithm for codes that are constructed from finite geometries is introduced. The application of this decoding algorithm to Euclidean geometry (EG) and projective geometry (PG) codes is further discussed. It is shown that these codes can be orthogonalized in less than or equal to three steps. Thus, these codes are majority-logic decodable in no more than three steps. Our results greatly reduce the decoding complexity of EG and PG codes in most cases. They should make these codes very attractive for practical use in error-control systems.
TL;DR: In this correspondence a complete decoding algorithm for double-error-correcting binary BCH codes of length n = 2^m - 1 is introduced, based on the step-by-step decoding algorithm introduced by Prange and the decoding algorithms introduced by Meggitt, which makes use of the cyclic properties of the code.
Abstract: In this correspondence a complete decoding algorithm for double-error-correcting binary BCH codes of length n = 2^m - 1 is introduced. It corrects all patterns of one and two errors and all patterns of three errors that belong to cosets that have a coset leader of weight three. This algorithm is based on the step-by-step decoding algorithm introduced by Prange and the decoding algorithm introduced by Meggitt, which makes use of the cyclic properties of the code. A comparison between this method and previously existing ones is also given.
TL;DR: In this paper a simple coding scheme utilizing both sequential and algebraic coding is proposed, and bounds on its performance are derived using theoretical bounds on the performance of sequential decoding.
Abstract: In this paper a simple coding scheme utilizing both sequential and algebraic coding is proposed, and bounds on its performance are derived using theoretical bounds on the performance of sequential decoding. These bounds are compared with bounds on a similar, though more complex, scheme proposed by Falconer [2]. Except for sequential rates in a range strictly above R_{comp}, the bounds on the present scheme are shown to be superior.
TL;DR: Applying these decoding algorithms to known classes of maximum-distance separable linear codes, the amount of hardware required for implementation is only a small fraction of those required by the existing decoding algorithms.
Abstract: In this paper, some properties of maximum-distance separable linear codes are presented. Based on these properties, a decoding algorithm for correcting random errors is established. A simpler decoding algorithm for correcting burst errors is also given. Applying these decoding algorithms to known classes of maximum-distance separable linear codes, the amount of hardware required for implementation is only a small fraction of those required by the existing decoding algorithms.
TL;DR: A suboptimum decision-directed block decoder for a binary symmetric channel that makes use of past decoding decisions to update its estimate of the channel's initially unknown crossover probability is considered.
Abstract: We consider a suboptimum decision-directed block decoder for a binary symmetric channel that makes use of past decoding decisions to update its estimate of the channel's initially unknown crossover probability. The decoder has a threshold list decoding rule that uses the current estimated crossover probability. The estimate is updated by means of a stochastic approximation algorithm. It is shown to converge toward the true crossover probability with a bias that decreases exponentially with the code's block length, provided it never "runs away" toward zero after dropping below a certain critical value. The probability that this runaway phenomenon ever occurs is bounded by an expression that is exponentially decreasing in the code's block length and in the weight assigned to the initial estimate.
TL;DR: A new algorithm is given for the decoding of double-error-correcting binary b.h.c. codes that can be rather simply implemented and is particularly suitable for parallel implementation.
Abstract: A new algorithm is given for the decoding of double-error-correcting binary b.c.h. codes. It can be rather simply implemented and is particularly suitable for parallel implementation.
TL;DR: A new lower bound on definite decoding minimum distance for the class of systematic binary periodic convolutional codes is presented, which is everywhere stronger than Wagner's bound and has the same form as the bound obtained by Massey.
Abstract: A new lower bound on definite decoding minimum distance for the class of systematic binary periodic convolutional codes is presented. The bound is everywhere stronger than Wagner's bound and has the same form as the bound obtained by Massey for the class of systematic binary fixed convolutional codes. The bound is also shown to apply to a specific subclass of simply implemented periodic codes for which Wagner's bound also holds.