Locally recoverable codes from towers of function fields

TL;DR

This paper constructs sequences of locally recoverable algebraic geometry codes from towers of function fields, providing bounds for their parameters and analyzing the optimality of the codes over finite fields.

Contribution

It introduces new locally recoverable codes from towers of function fields and establishes bounds, including a sharp bound for the Garcia-Stichtenoth tower over finite fields.

Findings

Bound for code parameters is sharp for the first code in the sequence.

Detailed analysis of subsequent codes based on rational place distribution.

Construction of locally recoverable codes with provable parameter bounds.

Abstract

In this work we construct sequences of locally recoverable AG codes arising from a tower of function fields and give bound for the parameters of the obtained codes. In a particular case of a tower over $\mathbb{F}_{q^2}$ for any odd $q$, defined by Garcia and Stichtenoth in [GS2007], we show that the bound is sharp for the first code in the sequence, and we include a detailed analysis for the following codes in the sequence based on the distribution of rational places that split completely in the considered function field extension.