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—We consider the problem of fast computation of the Fourier transform over a finite field by decomposing an arbitrary polynomial into a sum of linearized polynomials. Examples of algorithms for the Fourier transform with complexity less than that of the best known analogs are given.

Application of the cyclotomic Fast Fourier Transform 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… (More)

In this paper we consider the problem of computing the Fast Fourier Transform of a polynomial over finite fields. The polynomial is decomposed into a sum of linearized polynomials allowing one to use fast evaluation algorithms. An example of the FFT algorithm with the complexity lower than the best one known to the authors is provided.

—In this letter, 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.

A simple and natural Gao algorithm for decoding algebraic codes is described. Its relation to the Welch-Berlekamp and Euclidean algorithms is given. I. INTRODUCTION In the recent paper, Gao [1] described a simple and natural algorithm for decoding algebraic codes in the class of algorithms decoding up to the designed error-correcting capability. The… (More)

In this paper, following [1, 2, 3, 4, 5, 6, 7] we consider the relations between well-known Fourier transform algorithms.

In this paper we suggest a hybrid method for finding toots of error locator polynomials. We first review a fast version of the Chien search algorithm based on the decomposition of the error locator polynomial into a sum of multiples of affine polynomials. We then propose to combine it with modified analytical methods for solution of polynomials of small… (More)