Multiple-error-correcting Reed-Solomon or BCH codes in GF(2b) can be used for correction of multiple burst errors in binary data. However, the relatively long time required for decoding multiple errors has been among the main objections to applying these schemes to high-performance computer products. In this paper, a decoding procedure is developed for on-the-fly correction of multiple symbol (byte) errors in Reed-Solomon or BCH codes. A new decoder architecture expands the concept of Chien search of error locations into computation of error values as well, and creates a synchronous procedure for complete on-the-fly error correction of multiple byte errors. Forney's expression for error values is further simplified, which results in substantial economies in hardware and decoding time. All division operations are eliminated from the computation of the error-locator equation, and only one division operation is required in the computation of error values. The special cases of fewer errors are processed automatically, using the corresponding smaller set of syndromes through a single set of hardware. The resultant decoder implementation is well suited for LSI chip design with pipelined data flow. The implementation is illustrated with an example.