Linear-algebraic list decoding of folded Reed-Solomon codes

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 ε >; 0, the author and Rudra (2006, 08) presented an nO(1/ε) time algorithm to list decode appropriate folded RS codes of rate R from a fraction 1-R-ε 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 rootfinding step over extension fields (and indeed any mention of extension fields). The entire list decoding algorithm is linearalgebraic, 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 quadratic time. The theoretical drawback of folded RS codes are that both the decoding complexity and proven worst-case list-size bound are nΩ(1/ε). 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/ε2) which is quite close to the existential O(1/ε) bound (however, the decoding complexity remains nΩ(1/ε)). Our work highlights that constructing an explicit subspaceevasive subset that has small intersection with low-dimensional subspaces - an interesting problem in pseudorandomness in its own right - could lead to explicit codes with better listdecoding guarantees.