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IEEE Transactions on Information Theory

Efficient priority-first search maximum-likelihood soft-decision decoding of linear block codes

Yunghsiang Sam Han, C.R.P. Hartmann, Chih-Chieh Chen

Published Jan 1, 1993• 127 citations
Publisher

Abstract

The authors present a novel and efficient maximum-likelihood soft-decision decoding algorithm for linear block codes. The approach used here converts the decoding problem into a search problem through a graph that is a trellis for an equivalent code of the transmitted code. A generalized Dijkstra's algorithm, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to reduce drastically the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. For example, for most real channels of the 35 000 samples tried, simulation results for the (128,64) binary extended BCH code show that the proposed decoding algorithm is fifteen orders of magnitude more efficient in time and in space than that proposed by Wolf (1978). Simulation results for the (104, 52) binary extended quadratic residue code are also given.>

DOI
10.1109/18.259636

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IEEE Journal on Selected Areas in Communications•Jan 1, 1998•100 citations
The authors present a novel and efficient maximum-likelihood soft-decision decoding algorithm for linear block codes. The approach used here converts the decoding problem into a search problem through a graph that is a trellis for an equivalent code of the transmitted code. A generalized Dijkstra's algorithm, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to reduce drastically the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. For example, for most real channels of the 35 000 samples tried, simulation results for the (128,64) binary extended BCH code show that the proposed decoding algorithm is fifteen orders of magnitude more efficient in time and in space than that proposed by Wolf (1978). Simulation results for the (104, 52) binary extended quadratic residue code are also given.>
An efficient maximum-likelihood soft-decision decoding algorithm for linear block codes using a generalized Dijkstra's algorithm was proposed by Han, Hartmann, and Chen (1993). We prove that this algorithm is efficient for most practical communication systems where the probability of error is less than 10/sup -3/ by finding an upper bound of the computational effort of the algorithm. A suboptimal decoding algorithm is also proposed. The performance of this suboptimal decoding algorithm is within 0.25 dB of the performance of an optimal decoding algorithm for the (104, 52) binary extended quadratic residue code, and within 0.5 dB of the optimal performance for the (128, 64) binary BCH code, respectively.
Based on the notion of supercodes, we propose a two-phase maximum-likelihood (ML) soft-decision decoding (tpMLSD) algorithm for binary linear block codes in this work. The first phase applies the priority-first search algorithm backwardly to a trellis derived from the parity-check matrix of the supercode of the linear block code. Using the information retained from the first phase, the second phase employs the priority-first search algorithm to the trellis corresponding to the linear block code itself, which guarantees to find the ML decision with a constant complexity per information bit at high signal-to-noise ratios (SNRs). Simulations on the extended BCH code of n = 64 and k = 24 show that the proposed two-phase scheme is an order of magnitude more efficient in average decoding complexity than the recursive ML decoding [1] when the SNR per information bit is 8 dB.
This dissertation deals with maximum-likelihood soft-decision decoding as well as suboptimal soft-decision decoding of linear block codes. We present three new decoding algorithms: $MA\sp*D,\ H\sp*,$ and Directed Search. The first algorithm, $MA\sp*D,$ is an instance of Algorithm $A\sp*$ that conducts heuristic search through a code tree of the transmitted code when the decoding problem is transformed into a problem of graph-search through a code tree. $MA\sp*D$ takes into consideration more properties of the code and is considerably more efficient than the original $A\sp*$ decoding algorithm presented by Han, Hartmann, and Chen in 1993. The second algorithm, $H\sp*,$ is a new decoding algorithm that determines the value of every component of a minimum-cost codeword by estimating the cost of the minimum-cost codeword, which has a fixed value at one of the k most reliable, linearly independent bit positions. The suboptimal version of this algorithm is combined with other decoding algorithms to reduce the search space during the decoding process. The third algorithm, Directed Search, is designed to enhance the performance of soft-decision decoding by searching in continuous space. This approach explores the search space between a given vector and the received vector and finds the codeword closest to the received vector in the space explored. We also present a novel and efficient hybrid decoding algorithm for (n, k) linear block codes. It hybridizes $MA\sp*D,\ H\sp*,$ and Directed Search algorithms, utilizing their strengths to make the decoding more efficient. Simulation results for this hybrid algorithm are presented for the (104,52) binary-extended quadratic residue code and for the (128,64), the (256, 131), and the (256, 139) binary-extended BCH codes. This hybrid algorithm can efficiently decode the (104,52) binary extended quadratic residue code and the (128,64) code for any signal-to-noise ratio and has near-optimal to optimal performance. Previously, no practical decoder could decode these two codes with such a performance for all signal-to-noise ratio ranges.
Based on the notion of supercodes, we propose a two-phase maximum-likelihood soft-decision decoding (tpMLSD) algorithm for binary linear block codes in this work. The first phase applies the Viterbi algorithm backwardly to a trellis derived from the parity-check matrix of the supercode of the linear block code. Using the information retained from the first phase, the second phase employs the priority-first search algorithm to the trellis corresponding to the linear block code itself, which guarantees finding the ML decision. Simulations on Reed-Muller codes show that the proposed two-phase scheme is an order of magnitude more efficient in average decoding complexity than the recursive maximum-likelihood decoding (RMLD) [1] when the signal-to-noise ratio per information bit is 4.5 dB.
Iterative decoding methods have gained interest, initiated by the results of the so-called "turbo" codes. The theoretical description of this decoding, however, seems to be difficult. Therefore, we study the iterative decoding of block codes. First, we discuss the iterative decoding algorithms developed by Gallager (1962), Battail et al. (1979), and Hagenauer et al. (1996). Based on their results, we propose a decoding algorithm which only uses parity check vectors of minimum weight. We give the relation of this iterative decoding to one-step majority-logic decoding, and interpret it as gradient optimization. It is shown that the used parity check set defines the region where the iterative decoding decides on a particular codeword. We make plausible that, in almost all cases, the iterative decoding converges to a codeword after some iterations. We derive a computationally efficient implementation using the minimal trellis representing the used parity check set. Simulations illustrate that our algorithm gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes. Reed-Muller codes, quadratic residue codes, double circulant codes, and cyclic finite geometry codes. We also present simulation results for product codes and parallel concatenated codes based on block codes.