A novel method for computation of the discrete Fourier transform over a finite field with reduced multiplicative complexity is described. If the number of multiplications is to be minimized, then the novel method for the finite field of even extension degree is the best known method of the discrete Fourier transform computation. A constructive method of constructing for a cyclic convolution over a finite field is introduced.

SUMMARYApplication of the cyclotomic fast Fourier transform (FFT) algorithm to the syndrome evaluation problem in classical Reed-Solomon decoders is described. A number of complexity reduction tricks is suggested. Application of the algorithm leads to significant reductions in the complexity of syndrome evaluation. Moreover, automatic generation of the program code implementing the described algorithm is possible.

In this paper we propose an improved algorithm for finding roots of polynomials over finite fields.This makes possible significant speed up of the decoding process of BCH, Reed-Solomon and some other error-correcting codes.

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

No abstract