A Function Field Approach Toward Good Polynomials for Further Results on Optimal LRC Codes

TL;DR

This paper explores the construction of good polynomials for optimal locally recoverable codes (LRCs) using algebraic function fields and Galois theory, advancing understanding of polynomial properties that enhance distributed storage systems.

Contribution

It develops a Galois-theoretic approach to characterize and analyze good polynomials for optimal LRC codes, extending previous methods and providing explicit polynomial examples.

Findings

Characterization of polynomials with minimal Galois groups

Properties of finite fields with specific Galois group sizes

Explicit formulas for the parameter $\

Abstract

Because of the recent applications to distributed storage systems, researchers have introduced a new class of block codes, i.e., locally recoverable (LRC) codes. LRC codes can recover information from erasure(s) by accessing a small number of erasure-free code symbols and increasing the efficiency of repair processes in large-scale distributed storage systems. In this context, Tamo and Barg first gave a breakthrough by cleverly introducing a good polynomial notion. Constructing good polynomials for locally recoverable codes achieving Singleton-type bound (called optimal codes) is challenging and has attracted significant attention in recent years. This article aims to increase our knowledge of good polynomials for optimal LRC codes. Using tools from algebraic function fields and Galois theory, we continue investigating those polynomials and studying them by developing the Galois theoretical approach initiated by Micheli in 2019. Specifically, we push further the study of a crucial parameter $\mathcal G(f)$ (of a given polynomial $f$), which measures how much a polynomial is "good" in the sense of LRC codes. We provide some characterizations of polynomials with minimal Galois groups and prove some properties of finite fields where polynomials exist with a specific size of Galois groups. We also present some explicit shapes of polynomials with small Galois groups. For some particular polynomials $f$, we give the exact formula of $\mathcal G(f)$.