Decoding of the (24, 12, 8) extended golay code up to four errors

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 E_b/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.