Lifted Multiplicity Codes

TL;DR

This paper studies lifted multiplicity codes, providing bounds on their rate and distance, analyzing their asymptotic behavior, and introducing a local self-correction algorithm to improve their practical utility.

Contribution

It offers new bounds on the rate and distance of lifted multiplicity codes, analyzes their asymptotic monomial behavior, and presents a local self-correction method.

Findings

Lifted multiplicity codes can outperform previous codes in redundancy and disjoint recovery sets.

The asymptotic fraction of certain monomials in these codes is characterized for fixed variables and large alphabet.

A local self-correction algorithm is developed for these codes.

Abstract

Lifted Reed-Solomon codes and multiplicity codes are two classes of evaluation codes that allow for the design of high-rate codes that can recover every codeword or information symbol from many disjoint sets. Recently, the underlying approaches have been combined to construct lifted bi-variate multiplicity codes, that can further improve on the rate. We continue the study of these codes by providing lower bounds on the rate and distance for lifted multiplicity codes obtained from polynomials in an arbitrary number of variables. Specifically, we investigate a subcode of a lifted multiplicity code formed by the linear span of $m$-variate monomials whose restriction to an arbitrary line in $\mathbb{F}_q^m$ is equivalent to a low-degree uni-variate polynomial. We find the tight asymptotic behavior of the fraction of such monomials when the number of variables $m$ is fixed and the alphabet size $q=2^\ell$ is large. For some parameter regimes, lifted multiplicity codes are then shown to have a better trade-off between redundancy and the number of disjoint recovering sets for every codeword or information symbol than previously known constructions. Additionally, we present a local self-correction algorithm for lifted multiplicity codes.