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International MultiConference of Engineers and Computer Scientists

Soft-decoding of the (23, 12, 7) Binary Golay Code

Wen-Ku Su, Pei-Yu Shih, Tsung‐Ching Lin, Trieu‐Kien Truong

Published Mar 1, 2008• 1 citation
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Abstract

Abstract—Since the (23, 12, 7) binary Golay code is a perfect code, a weight-4 error occurred is always decoded as a weight-3 error pattern by a hard decoding method. In this paper, an efficient soft-decision decoder of the (23, 12, 7) binary Golay code up to the four errors is proposed. All probable patterns of occurred weight-4 error, which are always decoded to the same weight-3 error pattern, are determined from the look-up table of weight-7 codewords. And the most possible error pattern of weight-4 or weight-3 will be obtained by estimating the emblematic probability values of all probable patterns. The simulation result of this decoder in additive white Gaussian noise (AWGN) shows that at least 93 % and 99 % of weight-4 error patterns occurred are corrected if a bit-energy to noise-spectral-density ratios (E b/N 0) are greater than 3 dB and 6 dB, respectively, and at least 96 % of weight-3 error patterns occurred are corrected for any dB. I.

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International Journal of Systems Science•Apr 1, 1995•11 citations
Abstract—Since the (23, 12, 7) binary Golay code is a perfect code, a weight-4 error occurred is always decoded as a weight-3 error pattern by a hard decoding method. In this paper, an efficient soft-decision decoder of the (23, 12, 7) binary Golay code up to the four errors is proposed. All probable patterns of occurred weight-4 error, which are always decoded to the same weight-3 error pattern, are determined from the look-up table of weight-7 codewords. And the most possible error pattern of weight-4 or weight-3 will be obtained by estimating the emblematic probability values of all probable patterns. The simulation result of this decoder in additive white Gaussian noise (AWGN) shows that at least 93 % and 99 % of weight-4 error patterns occurred are corrected if a bit-energy to noise-spectral-density ratios (E b/N 0) are greater than 3 dB and 6 dB, respectively, and at least 96 % of weight-3 error patterns occurred are corrected for any dB. I.
In this paper, an efficient soft-decision decoder of the (24, 12, 8) binary Golay code for the four errors, composed of two stages: a conventional hard decoder up to three errors and the soft decoding for four errors, is proposed. All probable patterns of occurred weight-4 error, which are always decoded to the same weight-3 error pattern, are determined from the lookup table of weight-7 codewords. And the most possible one will be obtained by estimating the emblematic probability values of all probable patterns. The simulation result of this decoder in additive white Gaussian noise (AWGN) shows that at least 92% and 99% of weight-4 error patterns occurred are corrected if a bit-energy to noise-spectral-density ratios (Eb/N0) are greater than 2 dB and 5 dB, respectively.
A new decoder is proposed to decode the (23, 12, 7) binary Golay-code up to five errors. It is based on the algorithm that can correct up to four errors for the (24, 12, 8) extended Golay-code proposed by Lin et al., thereby achieving the soft decoding in the real sense for the Golay-code. For a weight-2 or weight-3 error pattern decoded by the hard decoder for correcting up to three errors, one can find the corresponding 21 weight-4 or weight-5 error patterns and choose the one with the maximum emblematic probability value, which is defined as the product of individual bit-error probabilities corresponding to the non-zero locations of the error pattern as the ultimate choice. Finally, simulation results of this decoder over additive white Gaussian noise (AWGN) channels show that the proposed method provides 0.9 dB coding gain than that of Lin et al.'s algorithm at bit-error rate of 10−5.
A new decoder is proposed to decode the (24, 12, 8) binary extended Golay code up to four errors. It consists of the conventional hard decoder for correcting up to three errors, the detection algorithm for four errors and the soft decoding for four errors. For a weight-4 error in a received 24-bit word, Method 1 or 2 is developed to determine all six possible error patterns. The emblematic probability value of each error pattern is then defined as the product of four individual bit-error probabilities corresponding to the locations of the four errors. The most likely one among these six error patterns is obtained by choosing the maximum of the emblematic probability values of all possible error patterns. Finally, simulation results of this decoder in additive white Gaussian noise show that at least 93% and 99% of weight-4 error patterns that occur are corrected if the two Eb/N0 ratios are greater than 2 and 5 dB, respectively. Consequently, the proposed method can achieve a better percentage of successful decoding for four errors at variable signal-to-noise ratios than Lu et al.'s algorithm in software. However, the speed of the method is slower than Lu et al.'s algorithm.
In this paper, a new soft-decision decoder of the (24, 12, 8) binary extended Golay code up to six errors is proposed. First, by using the error pattern obtained from hard decoder, the method of determining possible error patterns is developed. The emblematic probability value of each error pattern is then defined as the product of the individual bit-error probabilities corresponding to the locations of the possible error patterns. The most likely one among these error patterns is obtained by choosing the maximum of the emblematic probability values of possible error patterns. Finally, simulation results in additive white Gaussian noise (AWGN) show that this decoder reduce the decoding complexity although it performs a slight loss of coding gain than the modified Chase's II algorithm proposed by Hackett.
In this paper, a new decoding algorithm of the extended binary Golay code is proposed. Based on the proposition developed in this paper, all probable patterns of occurred weight-4 error, which are always decoded to the same weight-3 error pattern, are determined from an iterative invert-one-bit scheme instead of the look-up table of weight-7 codewords. And the most possible one will be obtained by estimating the emblematic probability values of all probable patterns. The simulation result of this decoder in AWGN shows that more than 99% of four errors are corrected if Eb/N0 is greater than 5 dB.
Two new algorithms for decoding the (23, 12) binary Golay code are developed with channel measurement information. For a white gaussian noise channel, each of the two algorithms achieves about 0·8 dB of decoding performance over a conventional hard-decision decoding. Furthermore, one of the algorithms requires a very small amount of computation for decoding a received block, and the other is very suitable for hardware implementation owing to its simple and regular operation. The hardware decoder, based on the latter algorithm, is designed with a pipeline such that it decodes one bit of a received block in a single system clock cycle