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TitleUse of Gröbner bases to decode binary cyclic codes up to the trueminimum distance
Author(s) Xuemin Chen, Tor Helleseth, Irving S. Reed, Trieu-Kien Truong
TypeArticle in Journal
AbstractA general algebraic method for decoding all types of binary cyclic codes is presented. It is shown that such a method can correct t=[(d-1)/2] errors, where d is the true minimum distance of the given cyclic code. The key idea behind this decoding technique is a systematic application of the algorithmic procedures of Grobner bases to obtain the error-locator polynomial L(z). The discussion begins from a set of syndrome polynomials F and the ideal T(F) generated by F. It is proved here that the process of transforming F to the normalized reduced Grobner basis of I(F) with respect to the “purely lexicographical” ordering automatically converges to L(z). Furthermore, it is shown that L(z) can be derived from any normalized Grobner basis of I(F) with respect to any admissible total ordering. To illustrate this new approach, the procedures for decoding certain BCH codes and quadratic residue codes are demonstrated
JournalIEEE Transactions on Information Theory
Translation No
Refereed No