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We present a simple syndrome-based fast Chase decoding algorithm forReed--Solomon (RS) codes. Such an algorithm was initially presented by Wu (IEEETrans. IT, Jan. 2012), building on properties of the Berlekamp--Massey (BM)algorithm. Wu devised a fast polynomial-update algorithm to construct theerror-locator polynomial (ELP) as the solution of a certain linear-feedbackshift register (LFSR) synthesis problem. This results in a conceptuallycomplicated algorithm, divided into $8$ subtly different cases. Moreover, Wu'spolynomial-update algorithm is not immediately suitable for working withvectors of evaluations. Therefore, complicated modifications were required inorder to achieve a true "one-pass" Chase decoding algorithm, that is, a Chasedecoding 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-basedfast 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 LFSRsynthesis problem, it is much simpler to consider "the right" minimizationproblem over a module. The solution to this minimization problem is a simplepolynomial-update algorithm that avoids syndrome updates and works seamlesslywith vectors of evaluations. As a result, we obtain a conceptually simplealgorithm for one-pass Chase decoding of RS codes. Our algorithm is generalenough to work with any algorithm that finds a Groebner basis for the solutionmodule of the key equation as the initial algorithm (including the Euclideanalgorithm), and it is not tied only to the BM algorithm.

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