List-decoding algorithms for lifted codes

TL;DR

This paper introduces efficient list-decoding algorithms for lifted Reed-Solomon codes, a family of error-correcting codes with better rate and local-decoding properties than Reed-Muller codes, by leveraging a new polynomial degree lemma.

Contribution

The paper presents the first efficient list-decoding algorithms for lifted codes, utilizing a novel lemma relating their codewords to low degree univariate polynomials.

Findings

Algorithms achieve efficient list-decoding of lifted codes

Codewords are low degree polynomials over a large field

Lifted codes have improved rate and local-decoding capabilities

Abstract

Lifted Reed-Solomon codes are a natural affine-invariant family of error-correcting codes which generalize Reed-Muller codes. They were known to have efficient local-testing and local-decoding algorithms (comparable to the known algorithms for Reed-Muller codes), but with significantly better rate. We give efficient algorithms for list-decoding and local list-decoding of lifted codes. Our algorithms are based on a new technical lemma, which says that codewords of lifted codes are low degree polynomials when viewed as univariate polynomials over a big field (even though they may be very high degree when viewed as multivariate polynomials over a small field).