On decoding of both errors and erasures of a Reed-Solomon code using an inverse-free Berlekamp-Massey algorithm

In a previous article by Truong et al. (see ibid., vol.46, p.973-76, 1998), it was shown that an inverse-free Berlekamp-Massey (1968, 1969) algorithm can be generalized to find the error locator polynomial in a Reed-Solomon (RS) decoder for correcting errors as well as erasures. The basic idea of this procedure is the replacement of the initial condition of an inverse-free BM algorithm by the Forney (1965) syndromes. It is shown that the errata locator polynomial can be obtained directly by initializing an inverse-free BM algorithm with the erasure locator polynomial and the syndromes. An important ingredient of this new algorithm is a modified BM algorithm for computing the errata locator polynomial. As a consequence, the separate computation of the erasure locator polynomial and the Forney syndrome, needed in the decoder developed by Truong et al., are completely avoided in this modification of the BM algorithm. This modified algorithm requires fewer finite field addition and multiplication operations than the previous algorithm. Finally, the new decoding method was implemented on a computer using C++ language. It is shown in a simulation that the speed of this new decoder is faster than the decoder developed by Truong et al. An example using this program is given for an (255, 239) RS code for correcting errors and erasures with 2ν+s⩽10