Classification of good rational functions improves storage code design
Good rational functions up to degree six
Information Theory
Summary
Distributed storage systems rely on special codes to keep data safe and quickly recover lost parts. The authors study good rational functions, which help build these codes more efficiently than older methods using polynomials. They focus on understanding these functions up to degree six in a specific mathematical setting, providing exact ways to calculate how well the codes will perform. This means they can design better and longer codes for storing data reliably.
What this means in practice
- •For distributed storage engineers: Create longer and more efficient locally recoverable codes for data storage systems using new constructions based on classified rational functions of degree up to six.
- •For error correction developers: Calculate exact code parameters quickly by leveraging the linear recurring sequence property of totally split places in extension fields, improving optimal code design.
Authors
Anuvrat Jaindungarwal, Divyesh Vaghasiya
Abstract
Locally recoverable codes (LRCs) are fundamental to the architecture of modern distributed storage systems. The Tamo-Barg construction of optimal LRCs was revolutionized by the introduction of good polynomials, which group evaluation points into local recovery sets. However, polynomial constructions are constrained by their inherently large Galois groups, which limit the number of totally split rational places. Recent advances have generalized this paradigm to good rational functions, which leverage the geometry of the projective line to achieve minimal Galois groups and subsequently codes with greater length. In this paper, we complete the classification of good rational functions of degree up to $6$ in characteristic $2$. By embedding the problem into algebraic function field theory, we explicitly determine the minimal arithmetic and geometric Galois groups for these low degrees. Furthermore, we move beyond asymptotic Hasse-Weil estimates by proving that the exact number of totally split places over extension fields obeys a linear recurring sequence (LRS). This gives an efficient method for computing the exact lengths and dimensions of the corresponding optimal LRCs and, in many cases, yields parameters beyond those obtainable from good polynomials.