Quadratic-Curve-Lifted Reed-Solomon Codes

TL;DR

This paper introduces quadratic-curve-lifted Reed-Solomon (QC-LRS) codes, analyzing their structure, bounds on dimension, rate, minimum distance, and demonstrating superior local recovery performance compared to lifted Reed-Solomon codes for short lengths.

Contribution

The paper defines QC-LRS codes, establishes their theoretical properties, and compares their local recovery capabilities with existing lifted Reed-Solomon codes.

Findings

Asymptotic rate of QC-LRS codes is approximately 1 - Θ(q/r)^-0.2284.

QC-LRS codes outperform lifted Reed-Solomon codes in local recovery for short lengths.

Bounds on the minimum distance of QC-LRS codes are analytically derived.

Abstract

Lifted codes are a class of evaluation codes attracting more attention due to good locality and intermediate availability. In this work we introduce and study quadratic-curve-lifted Reed-Solomon (QC-LRS) codes, where the codeword symbols whose coordinates are on a quadratic curve form a codeword of a Reed-Solomon code. We first develop a necessary and sufficient condition on the monomials which form a basis the code. Based on the condition, we give upper and lower bounds on the dimension and show that the asymptotic rate of a QC-LRS code over $\mathbb{F}_q$ with local redundancy $r$ is $1-\Theta(q/r)^{-0.2284}$. Moreover, we provide analytical results on the minimum distance of this class of codes and compare QC-LRS codes with lifted Reed-Solomon codes by simulations in terms of the local recovery capability against erasures. For short lengths, QC-LRS codes have better performance in local recovery for erasures than LRS codes of the same dimension.