A method for operating a Reed-Solomon encoder circuit, comprising: receiving a message having data symbols with the Reed-Solomon encoder circuit; computing a partial syndrome vector (430, 450) by multiplying the data symbols with a first matrix S u using a multiplier in the Reed-Solomon encoder circuit; with the Reed-Solomon encoder circuit, computing parity check symbols (350, 560, 630) by solving a system of linear equations that includes the partial syndrome vector (430, 450) and a second matrix S d; and with the Reed-Solomon encoder circuit, generating a Reed-Solomon code word (230, 640) by combining the data symbols and the parity check symbols (350, 560, 630), wherein the first matrix S u is defined as: S u = α t − 1 n − 1 ⋯ α n − 1, α 0 ⋮ ⋱ ⋮ α t − 1 n − k ⋯ α n − k, α 0, and the second matrix S d is defined as: S d = α t − 1 2 ⋯ α t − 1, α 0 ⋮ ⋱ ⋮ a 0 ⋯ a 0, a 0, wherein the second matrix S d is decomposed into a lower triangular matrix and an upper triangular matrix, and solving the system of linear equations further comprises: computing a forward vector in a finite field by performing a forward substitution based on the partial syndrome vector (430, 450) and multiplying the lower triangular matrix with the forward vector, wherein the Reed-Solomon code word has n symbols with n>k and includes t=n-k parity check symbols, the first matrix S u includes the first k rows of a parity check matrix of the Reed-Solomon code word, and the second matrix S d includes the last (n-k) rows of the parity check matrix of the Reed-Solomon code word, wherein the parity check matrix of the Reed-Solomon code word has n rows and t = (n-k) columns. receiving a message having data symbols with the Reed-Solomon encoder circuit; computing a partial syndrome vector (430, 450) by multiplying the data symbols with a first matrix S u using a multiplier in the Reed-Solomon encoder circuit; with the Reed-Solomon encoder circuit, computing parity check symbols (350, 560, 630) by solving a system of linear equations that includes the partial syndrome vector (430, 450) and a second matrix S d; and with the Reed-Solomon encoder circuit, generating a Reed-Solomon code word (230, 640) by combining the data symbols and the parity check symbols (350, 560, 630), wherein the first matrix S u is defined as: S u = α t − 1 n − 1 ⋯ α n − 1, α 0 ⋮ ⋱ ⋮ α t − 1 n − k ⋯ α n − k, α 0, and the second matrix S d is defined as: S d = α t − 1 2 ⋯ α t − 1, α 0 ⋮ ⋱ ⋮ a 0 ⋯ a 0, a 0, wherein the second matrix S d is decomposed into a lower triangular matrix and an upper triangular matrix, and solving the system of linear equations further comprises: computing a forward vector in a finite field by performing a forward substitution based on the partial syndrome vector (430, 450) and multiplying the lower triangular matrix with the forward vector, wherein the Reed-Solomon code word has n symbols with n>k and includes t=n-k parity check symbols, the first matrix S u includes the first k rows of a parity check matrix of the Reed-Solomon code word, and the second matrix S d includes the last (n-k) rows of the parity check matrix of the Reed-Solomon code word, wherein the parity check matrix of the Reed-Solomon code word has n rows and t = (n-k) columns. computing a forward vector in a finite field by performing a forward substitution based on the partial syndrome vector (430, 450) and multiplying the lower triangular matrix with the forward vector, wherein the Reed-Solomon code word has n symbols with n>k and includes t=n-k parity check symbols, the first matrix S u includes the first k rows of a parity check matrix of the Reed-Solomon code word, and the second matrix S d includes the last (n-k) rows of the parity check matrix of the Reed-Solomon code word, wherein the parity check matrix of the Reed-Solomon code word has n rows and t = (n-k) columns.