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Electronics and Communications in Japan (Part III Fundamental Electronic Science)

Variable threshold soft decision decoding

Kazuhiko Yamaguchi, Hiroshi Iizuka, Eiji Nomura, Hideki Imai

Published Jan 1, 1989• 7 citations
Publisher

Abstract

Abstract Soft decision decoding is a decoding method which tries to improve the error‐correcting capability by utilizing the information concerning the reliability of the received symbol. One of the soft decision decoding methods for practically useful block code, is APP (a posteriori probability) decoding applicable to the majority‐logic de‐codable code. In APP decoding, the error is decided for each symbol as in majority‐logic decoding. This paper proposes a new decoding method (variable threshold APP decoding), in which the error‐correcting capability is improved by controlling the order of error decisions in APP decoding. the performance is evaluated by simulation. As a result, it is shown that the variable threshold APP decoding exhibits better characteristics than APP decoding and approximate APP decoding both in bit error rate and block error rate. When, for example, the proposed decoding is applied to (73, 45) difference set cyclic code, a coding gain of 1.0 dB is obtained compared with the approximate APP decoding. In the proposed decoding method, the time required for decoding is increased, while the hardware scale is almost the same as that of the approximate APP decoding.

DOI
10.1002/ecjc.4430720907

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IEEE Transactions on Information Theory•Jan 1, 2000•31 citations
Abstract Soft decision decoding is a decoding method which tries to improve the error‐correcting capability by utilizing the information concerning the reliability of the received symbol. One of the soft decision decoding methods for practically useful block code, is APP (a posteriori probability) decoding applicable to the majority‐logic de‐codable code. In APP decoding, the error is decided for each symbol as in majority‐logic decoding. This paper proposes a new decoding method (variable threshold APP decoding), in which the error‐correcting capability is improved by controlling the order of error decisions in APP decoding. the performance is evaluated by simulation. As a result, it is shown that the variable threshold APP decoding exhibits better characteristics than APP decoding and approximate APP decoding both in bit error rate and block error rate. When, for example, the proposed decoding is applied to (73, 45) difference set cyclic code, a coding gain of 1.0 dB is obtained compared with the approximate APP decoding. In the proposed decoding method, the time required for decoding is increased, while the hardware scale is almost the same as that of the approximate APP decoding.
We present a new soft decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of all received symbols are recalculated into the reliabilities of the parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to the more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft decision algorithm outperforms its conventional hard decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output bit error rate /spl epsi/<1/2.
In this paper, an iterative soft-decision (SD) decoding algorithm for cyclic codes based on extended parity-check equations is developed. The algorithm does not necessarily utilize the algebraic properties of the code, but operates on transforming the systematic parity-check matrix using the soft reliability information matrix obtained from the received vector. Results show a significant performance gain when compared with the hard decision Berlekamp-Massey(B-M) and belief propagation (BP) algorithms, but present a similar bit error rate (BER) performance when compared to the adaptive belief propagation (ABP) algorithm. An important feature of the decoder is that it functions within a practical decoding time complexity, and can be generally implemented for the class of linear block codes.
We present novel decoding schemes for hard and soft decision decoding of block codes using the minimal weight codewords of the dual code. The decoding schemes will be described for cyclic codes where polynomials can be used, however, the modification for non-cyclic codes is possible and straight forward. The hard decision decoding calculates syndrome polynomials which are the product of the received polynomial with dual codewords. Proper cyclic shifts of these syndrome polynomials are obtained and the non-zero positions are counted componentwise for these shifts. The values of this counting are a reliability measure and can be used for locating the error and also the non-error positions. This reliability measure is the basis for various variants of hard decision decoding algorithms. Decoding schemes with iterative error reduction are possible as well as information set decoding using the inherent reliability information of the measure even if there is no reliability information from the channel. Further, we will show how reliability information from the channel can be included in order to obtain soft decision decoding schemes. We derive the relation between bit flipping, believe propagation, and majority logic decoding to the novel schemes. As examples to illustrate the functioning we use BCH and Reed-Muller codes as examples for binary codes, and RS codes for non-binary codes. Besides the Plotkin construction we recall a known result that Reed-Muller codes punctured by one position are cyclic and thus, are equivalent to special cases of BCH codes. Simulation results for hard and soft decision decoding will be given for several examples and compared with results from literature. Finally, we analyze the soft decision decoding of the Plotkin construction and derive that one of the two codes uses a 333 dB better channel (also known as channel polarization).
Binary cyclic codes achieve good error correction performance and allow the implementation of very simpleencoder and decoder circuits. Among them, BCH codesrepresent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that softdecision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on “extended parity-check matrices”, and show that such method is not suitable for high rates or long codes. We propose a new approach, based on “reduced parity-check matrices” and “spread parity-check matrices”, that can achieve better correction performance in many practical cases, without increasing the complexity.
This paper presents an approach to soft-decision decoding for a (n, k) binary linear block code. This approach achieves near-optimal to optimal performance for any signal-to-noise ratio value. The proposed decoding is based on the reordering of the received symbols according to their reliability measure and can be applied to any binary linear code. The basic idea is to generate a sequence of codewords such that there will be a high probability that the next codeword is closer to the received vector. In this algorithm the information provided by the soft-decisions is used to generate information sets utilized in decoding. A fast procedure checks whether a given set of k positions is an information set. A sequence of codewords is reconstructed using a sequence of information sets. Each codeword reconstructed is compared with the received word and the closest one is selected. Hence, the decoding algorithm is an iterative procedure designed to enhance the performance of the usual soft-decision decoding, since with each iteration of the procedure, real vectors successively closer to the received vector are examined. A high performance stop rule is utilized to reduce the number of generated information sets, stopping the decoding process when a codeword is closest to the received sequence.
We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM(r,m). First, the reliabilities of 2/sup m/ transmitted symbols are recalculated into the reliabilities of 2/sup m-r/ parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM(r,m), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log/sub 10/(/spl pi//2)/spl ap/2 dB at any given output error probability. For fixed code rate R and m/spl rarr//spl infin/, our algorithm increases almost 2/sup r/2/ times the correcting capability of soft-decision bounded distance decoding.