A decoder includes a syndrome generator for receiving a codeword and generating at least two syndromes based on the codeword, an error location polynomial generator for generating an error-location polynomial based on the syndromes, an error location determiner for determining at least one error location based on the error-location polynomial, and an error corrector for correcting the codeword based on the one error location. The error location polynomial generator includes a logic for receiving the syndromes and generating a combination of the syndromes as a combination of coefficients of the error-location polynomial, and a key equation solver for generating the error-location polynomial based on the combination of the coefficients and finding at least one root of the error-location polynomial. The error location determiner determines the error location based on a combination of the root and one of the syndromes.