A Groebner-bases approach to syndrome-based fast Chase decoding of Reed--Solomon codes

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), cleverly 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 Groebner 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" module minimization problem. 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 Groebner basis for the solution module of the key equation as the initial algorithm, and it is not tied only to the BM algorithm.