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IEEE Transactions on Information Theory

Maximum likelihood soft decoding of binary block codes and decoders for the Golay codes

Jakov Snyders, Yair Be'ery

Published Jan 1, 1989• 112 citations

Abstract

Maximum-likelihood soft-decision decoding of linear block codes is addressed. A binary multiple-check generalization of the Wagner rule is presented, and two methods for its implementation, one of which resembles the suboptimal Forney-Chase algorithms, are described. Besides efficient soft decoding of small codes, the generalized rule enables utilization of subspaces of a wide variety, thereby yielding maximum-likelihood decoders with substantially reduced computational complexity for some larger binary codes. More sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the (24, 12) Golay code, yielding a computational gain factor of about 2 with respect to previous methods. A ternary single-check version of the Wagner rule is applied for efficient soft decoding of the (12, 6) ternary Golay code.>

DOI: 10.1109/18.42214

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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.

Maximum likelihood (ML) decoding is the optimal error-correction method but is NP-hard in general, implying that there would be no polynomial-time ML decoding algorithm that capable of decoding arbitrary linear codes. Consequently, researchers have developed a variety of ML decoding algorithms with different characteristics so that they can be used in a complementary manner. Although many modern codes such as Turbo, LDPC, and Polar codes, can be efficiently decoded without ML algorithms, there remains continual demand for developing new ML decoding algorithms and evaluating ML decoding performance in various contexts. In this study, we propose an ML decoding algorithm applicable to general non-binary linear codes, formulated as a generalization of the adaptive and recursive ML decoding algorithm previously developed for binary codes. The proposed algorithm is implemented and applied to Generalized Reed-Muller codes, and its computational complexity of soft-decision ML decoding is discussed and compared with the complexity of the Viterbi decoder, in terms of the number of addition-equivalent operations, to demonstrate the efficiency of the approach. The substantial reduction in decoding complexity enables us performing practical decoding simulation of some classes of Generalized Reed-Muller codes. This paper also shows numerical evaluation results of the error-correcting capabilities of the codes, providing insights into their performance under optimal decoding conditions.