Algebraic decoding beyond eBCH of some binary cyclic codes, when e>eBCH

For a number of binary cyclic codes with e>e _BCH, algebraic algorithms are given to find the error locator polynomial. Thus, for these codes more errors can be corrected algebraically than by the Berlekamp-Massey algorithm. In some cases, all error patterns of weight up to e can be decoded; in other cases, only error patterns of weight up to e´ with e _BCH<e´⩽e can be decoded. The correctness of three of these algorithms is (partly) based on an exhaustive computer search; in all other cases, the algebraic proof is given in detail. It seems likely that many more cyclic codes can be decoded with these methods