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Journal of Information Science and Engineering, Vol. 23 No. 1, pp. 127-145 (January 2007)

Algebraic Decoding of Quadratic Residue Codes Using Berlekamp-Massey Algorithm*

Y. H. Chen, T. K. Truong, Y. Chang, C. D. Lee and S. H. Chen+
 School of Electrical and Information Engineering I-Shou University
Kaohsiung, 840 Taiwan
+Department of Computer Science and Information Engineering
Shu-Te University
Kaohsiung, 824 Taiwan

In this paper, an algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey algorithm. By a modification of the technique developed by He et al., one can express the unknown syndromes as functions of the known syndromes. The unknown syndromes are determined by an efficient algorithm also developed in this paper. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the Berlekamp-Massey algorithm. The decoding scheme, developed here, is easier to implement than the previous decoding algorithm developed for the Golay code and the (47, 24, 11) QR code. Moreover, it can be extended to decode all codes of the family of binary quadratic residue codes with irreducible generating polynomials.

Keywords: quadratic residue codes, unknown syndromes, known syndromes, Berlekamp- Massey algorithm, error-locator polynomial

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Received November 24, 2004; revised March 21, 2005; accepted March 31, 2005.
Communicated by Liang-Gee Chen.
* This work were supported by the National Science Council of Taiwan, R.O.C., under grants NSC 93-2213- E-214-001 and NSC 93-2215-M-214-003.