Algebraic decoding of (79,40,15) quadratic residue code using inverse-free Berlekamp-Massey algorithm

An algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey (BM) algorithm. By applying a technique developed by R. He et al. (see IEEE Trans. Inf. Theory, vol.47, p.1181-6, 2001), one can express unknown syndromes as functions of known syndromes. An efficient algorithm is also developed to determine the unknown syndromes. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the inverse-free BM algorithm. The new decoding scheme can be used to implement the (79,40,15) quadratic residue (QR) code which has not been treated so far. It is verified by a computer program that uses the C++ language.