An effective calculation of the Reed-Solomon code syndrome is proposed. The method is based on the use of the partial normalized cyclic convolutions in the partial inverse cyclotomic discrete Fourier transform. The method is the best of the known algorithms, in terms of multiplicative complexity.

A novel method for computing the discrete Fourier transform (DFT) over a finite field based on the Goertzel-Blahut algorithm is described. The novel method is currently the best one for computing the DFT over even extensions of the characteristic two finite field, in terms of multiplicative complexity.

A method to convert binary to pseudo-ternary sequences is proposed. The method keeps run-length constraints imposed on finite-length binary sequence and also provides a unit value of a channel transition. The resulting finite-length pseudo-ternary sequence has exponential sums falling within specified regions. This allows spectrum shaping. The average power spectral density of infinite sequences obtained by concatenating spectrally-constrained sequences is examined. Thus we obtain the pseudo-ternary spectrum shaped code, which generally yields higher capacity than the binary one.

A novel method for computation of the discrete Fourier transform over a finite field with reduced multiplicative complexity is described. The theorem about the multiplicative complexity coincidence of the Goertzel and cyclotomic algorithms is proved.

A simple and natural Gao algorithm for decoding algebraic codes is described. Its relation to the Welch-Berlekamp and Euclidean algorithms is given

We propose an improved algorithm for finding roots of polynomials over finite fields. This makes possible significant speedup of the decoding process of Bose-Chaudhuri-Hocquenghem, Reed-Solomon, and some other error-correcting codes.