# Constructive asymptotic bounds of locally repairable codes via function fields

## TL;DR

This paper introduces a new explicit method for constructing locally repairable codes using function fields, achieving improved asymptotic bounds without restrictions on locality or alphabet size, beneficial for distributed storage systems.

## Contribution

It provides the first explicit asymptotic construction of locally repairable codes over arbitrary finite fields from function field expansions, surpassing existing bounds.

## Key findings

- Achieves a Tsfasman-Vladut-Zink type bound for locally repairable codes.
- No constraints on locality and alphabet size in the construction.
- Exceeds Gilbert-Varshamov bound for large alphabet sizes.

## Abstract

Locally repairable codes have been investigated extensively in recent years due to practical applications in distributed and cloud storage systems. However, there are few asymptotical constructions of locally repairable codes in the literature. In this paper, we provide an explicit asymptotic construction of locally repairable codes over arbitrary finite fields from local expansions of functions at a rational place. This construction gives a Tsfasman-Vladut-Zink type bound for locally repairable codes. Its main advantage is that there are no constraints on both locality and alphabet size. Furthermore, we show that the Gilbert-Varshamov type bound on locally repairable codes over non-prime finite fields can be exceeded for sufficiently large alphabet size.