# Locality via Partially Lifted Codes

## TL;DR

This paper introduces a new class of error-correcting codes with adjustable locality properties, bridging the gap between two existing types, and provides constructions with higher rates using a novel variant of lifted codes.

## Contribution

The paper proposes a new code construction based on a variant of lifted codes that achieves higher rates for intermediate locality parameters s, bridging LCCs and LRCs.

## Key findings

- Constructed codes with higher rate for intermediate s values.
- Introduced a novel variant of lifted codes for code construction.
- Bridged the gap between LCCs and LRCs in locality properties.

## Abstract

In error-correcting codes, locality refers to several different ways of quantifying how easily a small amount of information can be recovered from encoded data. In this work, we study a notion of locality called the s-Disjoint-Repair-Group Property (s-DRGP). This notion can interpolate between two very different settings in coding theory: that of Locally Correctable Codes (LCCs) when s is large---a very strong guarantee---and Locally Recoverable Codes (LRCs) when s is small---a relatively weaker guarantee. This motivates the study of the s-DRGP for intermediate s, which is the focus of our paper. We construct codes in this parameter regime which have a higher rate than previously known codes. Our construction is based on a novel variant of the lifted codes of Guo, Kopparty and Sudan. Beyond the results on the s-DRGP, we hope that our construction is of independent interest, and will find uses elsewhere.