Algorithmic Improvements to List Decoding of Folded Reed-Solomon Codes

Abstract

Folded Reed-Solomon (FRS) codes are a well-studied family of codes, known for achieving list decoding capacity. In this work, we give improved deterministic and randomized algorithms for list decoding FRS codes of rate $R$ up to radius $1-R-\varepsilon$. We present a deterministic decoder that runs in near-linear time $\widetilde{O}_{\varepsilon}(n)$, improving upon the best-known runtime $n^{\Omega(1/\varepsilon)}$ for decoding FRS codes. Prior to our work, no capacity achieving code was known whose deterministic decoding could be done in time $\widetilde{O}_{\varepsilon}(n)$. We also present a randomized decoder that runs in fully polynomial time $\mathrm{poly}(1/\varepsilon) \cdot \widetilde{O}(n)$, improving the best-known runtime $\mathrm{exp}(1/\varepsilon)\cdot \widetilde{O}(n)$ for decoding FRS codes. Again, prior to our work, no capacity achieving code was known whose decoding time depended polynomially on $1/\varepsilon$. Our results are based on improved pruning procedures for finding the list of codewords inside a constant-dimensional affine subspace.