In this paper, some efficient methods of solving equations over Galois field GF(2m) are proposed. Using these algorithms, decoders for triple-and quadruple-error-correcting Bose–Chaudhuri–Hocquenghem (BCH) codes are shown. More- over, we propose a new method of making high-speed decoders for double-error-correcting/triple-error-detecting BCH or Reed- Solomon (RS) codes by adding a simple error-identifying circuit to a decoder for double-error-correcting codes. By incorporating ROM's (read only memory) in a decoder, the complex logic circuits are eliminated and then we can easily construct a high- speed decoder. We evaluate the complexity of the decoders and show that each of them can be accommodated in a single chip LSI.