Good rational functions up to degree six

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.