  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 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" 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 Groebner 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.
