Folded Reed-Solomon codes are an explicit family of codes that achieve the
optimal trade-off between rate and error-correction capability: specifically,
for any \eps > 0, the author and Rudra (2006,08) presented an n^{O(1/\eps)}
time algorithm to list decode appropriate folded RS codes of rate R from a
fraction 1-R-\eps of errors. The algorithm is based on multivariate
polynomial interpolation and root-finding over extension fields. It was noted
by Vadhan that interpolating a linear polynomial suffices if one settles for a
smaller decoding radius (but still enough for a statement of the above form).
Here we give a simple linear-algebra based analysis of this variant that
eliminates the need for the computationally expensive root-finding step over
extension fields (and indeed any mention of extension fields). The entire list
decoding algorithm is linear-algebraic, solving one linear system for the
interpolation step, and another linear system to find a small subspace of
candidate solutions. Except for the step of pruning this subspace, the
algorithm can be implemented to run in {\em quadratic} time. The theoretical
drawback of folded RS codes are that both the decoding complexity and proven
worst-case list-size bound are n^{\Omega(1/\eps)}. By combining the above
idea with a pseudorandom subset of all polynomials as messages, we get a Monte
Carlo construction achieving a list size bound of O(1/\eps^2) which is quite
close to the existential O(1/\eps) bound (however, the decoding complexity
remains n^{\Omega(1/\eps)}). Our work highlights that constructing an
explicit {\em subspace-evasive} subset that has small intersection with
low-dimensional subspaces could lead to explicit codes with better
list-decoding guarantees.Comment: 16 pages. Extended abstract in Proc. of IEEE Conference on
Computational Complexity (CCC), 201