Higher Hamming weights for locally recoverable codes on algebraic curves

TL;DR

This paper investigates the properties of locally recoverable codes on algebraic curves, introduces a new family based on Norm-Trace curves, and improves distance bounds for Hermitian LRC codes.

Contribution

It provides a generalized Hamming weight bound for algebraic curve-based LRC codes and proposes a novel family from Norm-Trace curves, enhancing code performance.

Findings

Derived a bound for generalized Hamming weights of LRC codes on algebraic curves.

Introduced a new family of algebraic geometric LRC codes from Norm-Trace curves.

Improved distance bounds for Hermitian LRC codes using properties of Hermitian codes.

Abstract

We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds of distance proposed in [1] for some Hermitian LRC codes. [1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic curves. arXiv preprint arXiv:1501.04904, 2015.