Beyond Polynomials: Optimal Locally Recoverable Codes from Good Rational Functions

2026-05-11

AI Summary

This work addresses the parameter limitations of polynomial-based constructions for locally recoverable codes (LRCs) by introducing the novel concept of “good rational functions,” which generalizes good polynomials to the realm of rational functions for the first time. Building upon this, the authors develop a unified and flexible framework for constructing optimal LRCs. By leveraging algebraic function fields and Galois theory to analyze the splitting behavior of rational functions, they explicitly construct several infinite families of optimal LRCs. The resulting codes significantly outperform the classical Tamo–Barg construction in key parameters such as locality, minimum distance, and code rate, thereby extending both the theoretical foundations and practical applicability of LRCs.

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.

Problem

• locally recoverable codes
• good rational functions
• optimal codes
• coding theory
• distributed storage

Innovation

• good rational functions
• locally recoverable codes
• algebraic function fields
• Galois theory
• optimal LRCs