Showing papers on "Sequential decoding published in 1978"

Efficient maximum likelihood decoding of linear block codes using a trellis

Abstract: It is shown that soft decision maximum likelihood decoding of any (n,k) linear block code over GF(q) can be accomplished using the Viterbi algorithm applied to a trellis with no more than q^{(n-k)} states. For cyclic codes, the trellis is periodic. When this technique is applied to the decoding of product codes, the number of states in the trellis can be much fewer than q^{n-k} . For a binary (n,n - 1) single parity check code, the Viterbi algorithm is equivalent to the Wagner decoding algorithm.

A bandwidth-efficient class of signal-space codes

Abstract: A new class of codes in signal space is presented, and their error and spectral properties are investigated. A constant-amplitude continuous-phase signal carries a coded sequence of linear-phase changes; the possible signal phases form a cylindrical trellis in phase and time. Simple codes using 4-16 phases, together with a Viterbi algorithm decoder, allow transmitter power savings of 2-4 dB over binary phase-shift keying in a narrower bandwidth. A method is given to compute the free distance, and the error rates of all the useful codes are given. A software-instrumented decoder is tested on a simulated Gaussian channel to determine multiple error patterns. The error parameter R_{o} is computed for a somewhat more general class of codes and is shown to increase rapidly when mere phases are employed. Finally, power spectral density curves are presented for several codes, which show that this type of coding does not increase the transmitted signal bandwidth.

The fast decoding of Reed-Solomon codes using Fermat theoretic transforms and continued fractions

Abstract: It is shown that Reed-Solomon (RS) codes can be decoded by using a fast Fourier transform (FFT) algorithm over finite fields GF(F_{n}) , where F_{n} is a Fermat prime, and continued fractions. This new transform decoding method is simpler than the standard method for RS codes. The computing time of this new decoding algorithm in software can be faster than the standard decoding method for RS codes.

An analysis of sequential decoding for specific time-invariant convolutional codes

Abstract: A new analysis of the computational effort and the error probability of sequential decoding is presented, which is based entirely on the distance properties of a particular convolutional code and employs no random-coding arguments. An upper bound on the computational distribution P(C_{t}>N_{t}) for a specific time-invariant code is derived, which decreases exponentially with the column distance of the code. It is proved that rapid column-distance growth minimizes the decoding effort and therefore also the probability of decoding failure or erasure. In an analogous way, the undetected error probability of sequential decoding with a particular fixed code is proved to decrease exponentially with the free distance and to increase linearly with the number of minimum free-weight codewords. This analysis proves that code construction for sequential decoding should maximize column-distance growth and free distance in order to guarantee fast decoding, a minimum erasure probability, and a low undetected error probability.

Syndrome decoding of binary-rate k/n convolutional codes

Abstract: A state-space approach to the syndrome decoding of binary rate k/n convolutional codes is described. State-space symmetries of a certain class of codes can be exploited to obtain a reduction in the exponent of growth of the decoder hardware. Aside from these savings it is felt that the state-space formalism developed has some unique intrinsic value.

Coding for Random-Access Memories

Abstract: A new decoding technique is presented for correcting errors due to bit-oriented hardware failures in parallel, random-access memories. It is shown that the resulting decoder compares favorably, both in complexity and in decoding delay, with currently implemented bit-switching techniques used for the same purpose.

Maximum-Likelihood Decoding of Binary Convolutional Codes on Band-Limited Satellite Channels

Abstract: Viterbi decoding of binary convolutional codes on bandlimited channels exhibiting intersymbol interference is considered, and a maximum likelihood sequence estimator algorithm is derived. This algorithm might be applied either to increase the allowable data rate for a fixed power transmitter or to reduce the required power for a fixed data rate. Upper and lower bounds on the bit error rate performance of several codes are found for selected values of the ratio of information rate to channel bandwidth, and results are compared against both conventional equalization techniques and the Shannon capacity limit. Results indicate that this algorithm can provide the power saving associated with low rate (highly redundant) codes without suffering the noise enhancement of linear equalization techniques.

A Simple Interleaver for Use with Viterbi Decoding

Abstract: An interleaver/de-interleaver can be used to improve the performance of a Viterbi decoder in the presence of bursty noise. This note describes the design and a method for testing block interleavers which can be used to combat the effects of periodic pulsed RFI. Simulations show that satisfactory decoder performance can be obtained over a range of RFI parameters.

Decoding of linear delta -decodable codes for a multiple- access channel (Corresp.)

Abstract: A scheme is presented for decoding linear \delta -decodable codes for the two-user noisy adder channel that exploits the linearity of the codes and corrects all patterns of \lfloor (\delta - 1)/2 \rfloor or fewer transmission errors.

Further results on decoding arithmetic residue codes (Corresp.)

Abstract: A decoding algorithm for a class of multiple error-correcting arithmetic residue codes can be made siginificantly more efficient.

An efficient minimum-distance decoding algorithm for convolutional error-correcting codes

Abstract: Minimum-distance decoding of convolutional codes has generally been considered impractical for other than relatively short constraint length codes, because of the exponential growth in complexity with increasing constraint length. The minimum-distance decoding algorithm proposed in the paper, however, uses a sequential decoding approach to avoid an exponential growth in complexity with increasing constraint length, and also utilises the distance and structural properties of convolutional codes to considerably reduce the amount of tree searching needed to find the minimum-distance path. In this way the algorithm achieves a complexity that does not grow exponentially with increasing constraint length, and is efficient for both long and short constraint length codes. The algorithm consists of two main processes. Firstly, a direct-mapping scheme, which automatically finds the minimum-distance path in a single mapping operation, is used to eliminate the need for all short back-up tree searches. Secondly, when a longer back-up search is required, an efficient tree-searching scheme is used to minimise the required search effort. The paper describes the complete algorithm and its theoretical basis, and examples of its operation are given.

Reducing the number of computations in stack decoding of convolutional codes by exploiting symmetries of the encoder

Abstract: This report describes a method of decoding of convolutional codes, where the decoding algorithm looks for an information-vector correction sequence, that must be added to the inverse of the received data vector sequence. This in contrast with the usual method, that tries to estimate the information vector sequence directly. Both methods are equally complex for a maximum likelihood (ML) decoder. In sequential decoding, with hard as well as with soft decisions, symmetries of the code can be exploited to reduce the number of computations and hence, the erasure probability. This will be illustrated by simulation results. The method is related to [1,2,31, where Schalkwijk et al. use state space symmetries of the syndrome former to obtain a reduction in the exponential rate of growth of the hardware of a Viterbi-like syndrome decoder. This research was partly supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).

Bidirectional search for convolutional codes

Abstract: In sequential decoding, the buffer overflow is often caused by the Paretian computational distribution problem. This troublesome distribution arises from the sequential decoding algorithm, not from the basic properties of convolutional codes. In this paper, we show that this problem could be removed by using a bidirectional search. In Section 2, we develop a bidirectional tree structure that is applicable to any given convolutional code. In Section 3, we fully interpret why a decoding sequence could recover to the correct path after accepting errors. Also, factors which would affect the length of recovery decoding error bursts are discussed. In Section 4, we first introduce four new properties of convolutional codes. Then we describe the general bidirectional search procedure and explain why and how this search could eliminate the recovery decoding errors. To clarify all the involvements and considerations of using the proposed scheme, examples are given for general reference. It is understood that the performance of a system using long codes depends on the decoding speed and the buffer capability of the decoder. Therefore, the new decoding approach presented in this paper could open a new direction toward overcoming the existing difficulties of using long convolutional codes.

Maximum likelihood syndrome decoding of linear block codes

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Simple proof of the continued fraction algorithm for decoding Reed-Solomon codes

Abstract: It was shown recently that BCH and RS codes can be implemented by Berlekamp's algorithm using continued fraction approximations. A simple transparent proof of Berlekamp's algorithm that uses such a development is given in this paper.

Addition to 'A Method for Decoding of Generalized Goppa Codes'

Abstract: In a previous correspondence, a decoding procedure which uses continued fractions and which is applicable to a wide class of algebraic codes including Goppa codes was presented The efficiency of this method is significantly increased

A decoding procedure for the Reed-Solomon codes

Abstract: A decoding procedure is described for the (n,k) t-error-correcting Reed-Solomon (RS) code, and an implementation of the (31,15) RS code for the I4-TENEX central system. This code can be used for error correction in large archival memory systems. The principal features of the decoder are a Galois field arithmetic unit implemented by microprogramming a microprocessor, and syndrome calculation by using the g(x) encoding shift register. Complete decoding of the (31,15) code is expected to take less than 500 microsecs. The syndrome calculation is performed by hardware using the encoding shift register and a modified Chien search. The error location polynomial is computed by using Lin's table, which is an interpretation of Berlekamp's iterative algorithm. The error location numbers are calculated by using the Chien search. Finally, the error values are computed by using Forney's method.

Sequential Operation of Communication Channels and the Capacity Using Variable-Length Codes

Abstract: We develop a mathematical theory of sequential operation of communication channels, and prove the coding theorem for the capacity using variable-length codes. The mathematical theory consists in constructing the sequential transfer probability function of a continuous-time channel relative to a stopping rule. It probabilistically describes the sequential output behavior of the channel in response to a given input sequence of functions. The mathematical definition of the capacity using variable-length codes is then given in terms of this probability function. The coding theorem for the capacity is established by first proving the information-stability theorem using a generalized ergodic theorem. It is shown that for some common channels the capacity using variable-length codes is numerically equal to the ordinary capacity using fixed-length codes.

Generalized permutations in convolutional codes

Abstract: The properties of convolutional codes and their automorphisms are thoroughly investigated. The structure of the automorphism group of a code is related to some easily obtained properties of a minimal encoder of the code. The class of convolutional operators is introduced and these operators are used to enumerate codes that are equivalent for use on a symmetric memoryless channel.

Minimum-distance decoding of binary convolutional codes

Abstract: The paper considers four novel decoders that come close to achieving minimum-distance decoding of convolutional codes, but without requiring nearly as much storage or nearly as many operations per received data symbol as does the Viterbi-algorithm decoder. The methods of operation of the decoders are first described and the results of computer simulation tests are then presented, comparing the tolerances of the decoders to additive white Gaussian noise with that of a Viterbi-algorithm decoder. Four different rate½ -binary convolutional codes and three different distance measures are used in the tests.

Decoding the binary Golay code with miracle octad generators (Corresp.)

Abstract: The recently introduced notion of a miracle octad generator can be viewed as a means of identifying a codeword of weight eight of the extended binary Golay code, given five of its nonzero positions. It is shown that this fact can be used as the basis of a new decoding algorithm for this code which decodes all the information positions simultaneously. The performance of this algorithm, as well as methods for its implementation, are considered.

Soft decision decoding of transmission codes

Abstract: Soft decision decoding has been successfully applied to convolutional and block-coding schemes. This latter suggests some alternative applications with particular attention being paid to transmission codes. Alternate mark inversion (a.m.i.) is used as an example where an effective signal/noise ratio improvement of up to 2.8 dB can be obtained.

Pseudostep orthogonalisation: an algorithm for improving Reed-Massey threshold codes

Abstract: An algorithm is presented which can be applied to Reed-Massey algorithm codes and utilises orthogonal and non-orthogonal check sums with a resulting improved performance. The algorithm is applied to a well-known class of convolutional threshold codes with a subsequent improvement in the nonbounded error-correcting capability.

An iterative algorithm for decoding block codes transmitted over a memoryless channel

Abstract: An algorithm was developed which optimally decodes a block code for minimum probability of symbol error in an iterative manner. The initial estimate is made by looking at each bit independently and is improved by considering bits related to it through the parity check equations. The dependent bits are considered in order of interesting probability of error. Since the computation proceeds in a systematic way with the bits having the greatest effect being used first, the algorithm approaches the optimum estimate after only a fraction of the parity check equations were used.

On Viterbi decoding for time-varying convolutional codes

Abstract: The Viterbi decoding technique has been extended for time-varying convolutional codes. The problem has been presented as a discrete-times, time-varying regulator control problem, and dynamic programming has been utilized to explain the decoding process.