Linear-Algebraic List Decoding for Variants of Reed–Solomon Codes

Folded Reed-Solomon (RS) codes are an explicit family of codes that achieve the optimal tradeoff between rate and list error-correction capability: specifically, for any ε > 0, Guruswami and Rudra presented an n^{O(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 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 quadratic time. We also consider a closely related family of codes, called (order m) derivative codes and defined over fields of large characteristic, which consist of the evaluations of f as well as its first m-1 formal derivatives at N distinct field elements. We show how our linear-algebraic methods for folded RS codes can be used to show that derivative codes can also achieve the above optimal tradeoff. The theoretical drawback of our analysis for folded RS codes and derivative codes is that both the decoding complexity and proven worst-case list-size bound are n^{Ω(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 decodin- complexity remains n^{Ω(1/ ε)}). Our work highlights that constructing an explicit subspace-evasive 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 list-decoding guarantees.