Hermitian-Lifted Codes

TL;DR

This paper introduces Hermitian-lifted codes, a new class of error-correcting codes constructed from the Hermitian curve that enable efficient local recovery of erasures with high availability and constant rate.

Contribution

The paper presents a novel construction of codes based on Hermitian curves using special monomials, enhancing local recovery capabilities in error correction.

Findings

Codes have high availability for local erasure recovery.

Codes achieve constant-bounded rate.

Recovery sets correspond to points on lines through a given point.

Abstract

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $\mathbb{F}_{q^2}$-rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point.