Curve-lifted codes for local recovery using lines

TL;DR

This paper introduces curve-lifted codes over arbitrary fields, inspired by Hermitian codes, focusing on local recovery and availability, with new families of codes based on norm-trace curves that offer improved asymptotic rates.

Contribution

It generalizes Hermitian-lifted codes to arbitrary fields and introduces norm-trace-lifted codes with enhanced parameters for local recovery.

Findings

Norm-trace-lifted codes are easier to define than Hermitian codes.

These codes achieve high availability and good asymptotic code rate.

The construction depends on intersection properties of curves and lines.

Abstract

In this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over $\mathbb{F}_{2^r}$. These codes are designed for locality and availability, and their particular parameters depend on the choice of curve and its properties. Due to the construction, the numbers of rational points of intersection between curves and lines play a key role. To demonstrate that and generate new families of locally recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted codes. In some cases, they are easier to define than their Hermitian counterparts and consequently have a better asymptotic bound on the code rate.