Fast algorithm for computing the roots of error locator polynomials up to degree 11 in Reed-Solomon decoders

Abstract

The central problem in the implementation of a Reed-Solomon code is finding the roots of the error locator polynomial. In 1967, Berlekamp et al. found an algorithm for finding the roots of an affine polynomial in GF(2 m ) that can be used to solve this problem. In this paper, it is shown that this Berlekamp-Rumsey-Solomon algorithm, together with the Chien-search method, makes possible a fast decoding algorithm in the standard-basis representation that is naturally suitable in a software implementation. Finally, simulation results for this fast algorithm are given. Index Terms-Berlekamp-Rumsey-Solomon algorithm, Chien search, error locator polynomial, -polynomial, Reed-Solomon code.

Finding roots of polynomials over finite fields

“…, α 254 were averaged over 100000 computations and shown in Table 1. Note that speedup rates for Truong, Jeng and Reed method are significantly lower than shown in [2]. This is caused by different implementation of multiplication operation used in our simulations.…”

“…Moreover, if some roots are located in the same cyclotomic coset, it is possible to eliminate them using Euclidean algorithm. In their recent paper [2] Truong, Jeng and Reed proposed a transformation which allows grouping of some summands of the polynomial of degree not higher than 11 into multiples of affine polynomials. Since affine polynomials can be easily evaluated using very small pre-computed tables, it is possible to speed up computations.…”

The generalized Goertzel algorithm and its parallel hardware implementation

“…By using the so-called linearized polynomials (see [9,10]), Berlekamp, Rumsey and Solomon [6] gave a fast algorithm for finding the roots of polynomials over finite fields. Truong et al [7] improved this algorithm for polynomials up to degree 11. Fedorenko and Trifonov proposed a further improved fast root-finding algorithm in [8].…”

“…For example, it is well-known that the most time-consuming part in the RS and BCH decoding is the determination of the error positions (the roots of the error locator polynomials, see [1][2][3]). Many authors have contributed to this problem and several fast root-finding algorithms were presented in [2,[4][5][6][7][8]. The most famous one is the Chien search [4] , which has been widely implemented in the RS and BCH decoders used in digital products (see [1][2][3]).…”

Efficient Procedure for Correcting Both Errors and Erasures of Reed-Solomon Codes

“…However, the decoders in [4] and [5] need to find the roots of the errata locator polynomial. A Chien search has been widely used for finding the roots of the error and errata polynomials [4], [5], and [10] despite the fact that faster methods for finding the roots of a polynomial have been developed [11], [12]. Notably, the computational requirements for finding the roots of an errata locator polynomial using a Chien search are n(µ +  -1) additions and n(µ +) multiplications over GF (2 m ), where n = 2 m − 1, which is very time-consuming, particularly for µ  .…”