# Lifted multiplicity codes and the disjoint repair group property

## TL;DR

This paper introduces lifted multiplicity codes, a generalization of lifted Reed Solomon codes, demonstrating they achieve superior redundancy-locality trade-offs and disjoint repair group properties, advancing error correction code efficiency.

## Contribution

It presents lifted multiplicity codes with improved redundancy and locality trade-offs, and provides a new analysis of lifted Reed Solomon codes via dual codes.

## Key findings

- Lifted multiplicity codes achieve redundancy $O(t^{0.585} \, \sqrt{N})$ with disjoint repair groups.
- They offer the best known trade-off for redundancy and locality for super-constant $t < \sqrt{N}$.
- Alternative analysis of lifted Reed Solomon codes using dual codes is provided.

## Abstract

Lifted Reed Solomon Codes (Guo, Kopparty, Sudan 2013) were introduced in the context of locally correctable and testable codes. They are multivariate polynomials whose restriction to any line is a codeword of a Reed-Solomon code. We consider a generalization of their construction, which we call lifted multiplicity codes. These are multivariate polynomial codes whose restriction to any line is a codeword of a multiplicity code (Kopparty, Saraf, Yekhanin 2014). We show that lifted multiplicity codes have a better trade-off between redundancy and a notion of locality called the $t$-disjoint-repair-group property than previously known constructions. More precisely, we show that lifted multiplicity codes with length $N$ and redundancy $O(t^{0.585} \sqrt{N})$ have the property that any symbol of a codeword can be reconstructed in $t$ different ways, each using a disjoint subset of the other coordinates. This gives the best known trade-off for this problem for any super-constant $t < \sqrt{N}$. We also give an alternative analysis of lifted Reed Solomon codes using dual codes, which may be of independent interest.