Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions

TL;DR

This paper introduces good rational functions as a generalization of good polynomials to construct optimal locally recoverable codes with improved parameters, leveraging algebraic function field and Galois theory.

Contribution

It extends the framework of LRC construction from good polynomials to good rational functions, providing explicit families that outperform previous methods.

Findings

Constructed explicit families of good rational functions.

Achieved infinite families of optimal LRCs with better parameters.

Extended Galois-theoretic framework for rational functions.

Abstract

Locally recoverable codes (LRCs) have emerged as fundamental objects in modern coding theory, primarily due to their pivotal role in distributed and cloud storage systems. A major breakthrough in their construction was achieved by Tamo and Barg, who introduced the notion of \emph{good polynomials} as a key structural ingredient. In this article, we propose a natural generalization of this paradigm by introducing the concept of \emph{good rational functions}. Building upon this extension, we develop a unified and flexible framework for constructing optimal LRCs. To quantify the quality of a rational function, we embed the problem into the rich context of algebraic function field theory and Galois theory. This perspective allows us to extend the Galois-theoretic framework originally developed by Micheli for good polynomials. In particular, we derive structural and quantitative results on the number of totally split rational places associated with rational functions. Furthermore, we construct explicit families of good rational functions that outperform all good polynomials of the same degree. As a consequence, we obtain infinite families of optimal LRCs with improved parameters compared to those arising from the classical Tamo-Barg construction. These results highlight the intrinsic strength of our approach.