Linear-Algebraic List Decoding of Folded Reed-Solomon Codes

Abstract

Folded Reed-Solomon codes are an explicit family of codes that achieve the optimal tradeoff between rate and error-correction capability: specifically, for any ε > 0, the author and Rudra (2006,08) 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 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 quadratic time.The theoretical drawback of folded RS codes are 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/ε 2 ) which is quite close to the existential O(1/ε) bound (however, the decoding 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.

A General Construction for 1-Round δ-RMT and (0, δ)-SMT

“…where (d) is because of the choice of s and (e) is because of the value of k in (10). Note that because s ≥ 1, we have …”

“…By appropriate choice of s and u, the code can reach 1 − R − radius and so asymptotically is optimal. Lemma 1 below, given in [10], gives the condition that needs to be satisfied by the two parameters and the number of errors to be corrected. Lemma 1.…”

“…In [10] a variant decoding for FRS-code is given in which the interpolation uses polynomial Q(X, Y 1 , · · · , Y s ) where degree of Y i is 1. This variant has a simpler exposition and allows a simpler way of choosing the code parameter, s and u.…”

A Model for Adversarial Wiretap Channels and its Applications

“…A decoding algorithm for FRS codes is given in Ref. [28], where decoding is by finding a solution to a set of linear equations. The output of the decoder is a list of codewords, where the list size is exponential in N.…”

“…3) Subspace Evasive Set: Subspace evasive sets [28] have been used to reduce the decoding list size of list decodable codes. A subset evasive set is a subset of F n q with the property that it has small intersection with every k-dimensional affine subspace.…”