A Gröbner-Bases Approach to Syndrome-Based Fast Chase Decoding of Reed-Solomon Codes - Shany, Yaron; Berman, Amit (IEEE transactions on information theory, 2022)
We present a simple syndrome-based fast Chase decoding algorithm for Reed-Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEE Trans. IT, Jan. 2012), building on properties of the Berlekamp-Massey (BM) algorithm. Wu devised a fast polynomial-update algorithm to construct the error-locator polynomial (ELP) as the solution of a certain linear-feedback shift register (LFSR) synthesis problem. This results in a conceptually complicated algorithm, divided into 8 subtly different cases. Moreover, Wu's polynomial-update algorithm is not immediately suitable for working with vectors of evaluations. Therefore, complicated modifications were required in order to achieve a true "one-pass" Chase decoding algorithm, that is, a Chase decoding algorithm requiring O(n) operations per modified coordinate, where n is the RS code length. The main result of the current paper is a conceptually simple syndrome-based fast Chase decoding of RS codes. Instead of developing a theory from scratch, we use the well-established theory of Gröbner bases for modules over \mathbb {F}_{q}[X] (where \mathbb {F}_{q} is the finite field of q elements, for q a prime power). The basic observation is that instead of Wu's LFSR synthesis problem, it is much simpler to consider "the right" minimization problem over a module . The solution to this minimization problem is a simple polynomial-update algorithm that avoids syndrome updates and works seamlessly with vectors of evaluations. As a result, we obtain a conceptually simple algorithm for one-pass Chase decoding of RS codes. Our algorithm is general enough to work with any algorithm that finds a Gröbner basis for the solution module of the key equation as the initial algorithm (including the Euclidean algorithm), and it is not tied only to the BM algorithm.