Lifted Projective Reed-Solomon Codes

TL;DR

This paper introduces and analyzes lifted projective Reed-Solomon codes, extending existing codes to projective spaces, deriving local correction algorithms, and demonstrating their practical implementation and parameters.

Contribution

It formalizes the definition of lifted projective Reed-Solomon codes, studies their parameters, and develops local correction algorithms with practical implementation.

Findings

Derived local correction algorithms from code structure.

Established links between lifted Reed-Solomon and projective codes via shortening and puncturing.

Provided practical implementation and parameter analysis.

Abstract

Lifted Reed-Solomon codes, introduced by Guo, Kopparty and Sudan in 2013, are known as one of the few families of high-rate locally correctable codes. They are built through the evaluation over the affine space of multivariate polynomials whose restriction along any affine line can be interpolated as a low degree univariate polynomial. In this work, we give a formal definition of their analogues over projective spaces, and we study some of their parameters and features. Local correcting algorithms are first derived from the very nature of these codes, generalizing the well-known local correcting algorithms for Reed-Muller codes. We also prove that the lifting of both Reed-Solomon and projective Reed-Solomon codes are deeply linked through shortening and puncturing operations. It leads to recursive formulae on their dimension and their monomial bases. We finally emphasize the practicality of lifted projective Reed-Solomon codes by computing their information sets and by providing an implementation of the codes and their local correcting algorithms.