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.

Binary quadratic residue (QR) codes, which have code rates greater than or equal to 1/2 and generally have large minimum distances, are among the best known codes. This paper considers a modified algebraic decoding algorithm for the (89,45,17) binary QR code that utilizes the Berlekamp-Massey algorithm. It identifies the primary unknown syndromes and provides methods to determine these on a case-by-case basis for any number of correctable errors. Numerical evaluation shows that the proposed algorithm significantly reduces at least 52% of decoding time for two or more 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 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.

Recently, an algebraic decoding algorithm suggested by Truong (2005) for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial. In this paper, based on the ideas of the algorithm mentioned above, an algebraic decoder for the (89, 45, 17) binary quadratic residue code, the last one not decoded yet of length less than 100 , is proposed. It was also verified theoretically for all error patterns within the error-correcting capacity of the code. Moreover, the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations.

Construction methods for self-dual codes are given. By using these methods some new extremal self-dual$[{66, 33, 12}]$and$[{68, 34, 12}]$codes are obtained.