Wen-Ku Su

Algebraic Decoding of the

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.

(89, 45, 17)

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.

Quadratic Residue Code

In this letter, the algebraic decoding algorithm of the (89, 45, 17) binary quadratic residue (QR) code proposed by Truong et al. is modified by using the efficient determination algorithm of the primary unknown syndromes. The correctness of the proposed decoding algorithm is verified by computer simulations and the use of two corollaries. Also, simulation results show that the CPU time of this algorithm is approximately 4 times faster than that of the previously mentioned decoding algorithm at least. Therefore, such a fast decoding algorithm can now be applied to achieve efficiently the reliability-based decoding for the (89, 45, 17) QR code. Finally, the performance of its algebraic soft-decision decoder expressed in terms of the bit-error probability versus Eb/N0 is given but not available in the literature.

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

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 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 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.

Fast Algebraic Decoding of the (89, 45, 17) Quadratic Residue Code

This paper is to develop a modified algorithm for decoding the (48, 24, 12) binary quadratic residue code up to six errors. The technique in this paper uses the algebraic decoding algorithm for the (47, 24, 11) quadratic residue code offered by Truong et al. to correct up to five errors. Then, the technique of detecting a six-error is utilized. Finally, the reliability-search algorithm, proposed by Dubney et al., is applied for correcting the six-error. The computer simulation of the new scheme shows that at least 86% and 96% of weight-6 error patterns occurred are corrected if the Eb/N0 ratios are greater than 3 dB and 6 dB, respectively.

On soft-decoding of the (24, 12, 8) Golay code

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.

Decoding of the (48, 24, 12) extended quadratic residue code up to six errors

This letter proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleasons theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distributions of binary (138, 69, 22) and (168, 84, 24) extended quadratic residue codes are given.

Modified algebraic decoding of the (89, 45, 17) binary quadratic residue code

This paper proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleasonpsilas theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distribution of binary (168, 84, 24) extended quadratic residue code is given.

A New Scheme to Determine the Weight Distributions of Binary Extended Quadratic Residue Codes

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.

On determination of the weight distribution of binary (168, 84, 24) extended quadratic residue code

An algebraic decoding of the (89, 45, 17) quadratic residue code suggested by Truong et al. (2008) has been designed that uses the inverse-free Berlekamp-Massey (BM) algorithm to determine the error-locator polynomial and applies a verification method to check whether the error pattern obtained by decoding algorithm is correct or not. In this paper, based on the ideas of the algorithm mentioned above, two decoding methods of the (31, 16, 7) binary quadratic residue code are proposed. Also, the comparison of the decoding complexity in terms of CPU time of these two methods and the conventional algebraic decoding method proposed by Reed et al. (1990) are given.