A NEW PROCEDURE FOR DECODING CYCLIC AND BCH CODES UP TO ACTUAL MINIMUM DISTANCE

This paper presents a new procedure for decoding cyclic and BCH codes up to their actual minimum distance. It generalizes the Peterson decod ing procedure and our recent procedure using nonrecurrent syndrome dep endence relations. For a code with actual minimum distance d to correc t up to t = [(d - 1) / 2] errors, the procedure requires a (2t + 1) X (2t + 1) syndrome matrix with known syndromes above the minor diagonal and unknown syndromes and their conjugates on the minor diagonal. In contrast to previous procedures, this procedure is primarily aimed at solving for the unknown syndromes instead of determining an error-loca tor polynomial. Decoding is then accomplished by determining the error vector as the inverse Fourier transform of the syndrome vector (S-0, S-1(...), S-n-1). We have shown that, with this procedure, all binary cyclic and BCH codes of length < 63 (with one exception) can be decode d up to their actual minimum distance. The procedure incorporates an e xtension of our fundamental iterative algorithm and a majority scheme for confirming the true values computed for the unknown syndromes. The complexity of this decoding procedure is O(n(3)).