A new procedure for decoding cyclic and BCH codes up to actual minimum distance

The paper presents a new procedure for decoding cyclic and BCH codes up to their actual minimum distance. It generalizes the Peterson decoding procedure and the procedure of Feng and Tzeng (1991) using nonrecurrent syndrome dependence relations. For a code with actual minimum distance d to correct up to t=[(d-1)/2] errors, the procedure requires a (2t+1)×(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-locator polynomial. Decoding is then accomplished by determining the error vector as the inverse Fourier transform of the syndrome vector (S₀, S₁, S_n-1). The authors show that with this procedure, all binary cyclic and BCH codes of length <63 (with one exception) can be decoded up to their actual minimum distance. The procedure incorporates an extension of their 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(n³)