Optimal Selection for Good Polynomials of Degree up to Five

TL;DR

This paper classifies and constructs good polynomials of degree up to five, providing bounds and estimates for their maximal sets, which are crucial for designing optimal Locally Recoverable Codes.

Contribution

It offers a complete classification of good polynomials up to degree five, explicit bounds on their maximal sets, and methods to construct optimal polynomials for LRC applications.

Findings

Explicit bounds on the maximal number of constant sets for degrees up to 4

Construction methods for achieving bounds in degree 5

Computational evidence showing bounds are close to actual values

Abstract

Good polynomials are the fundamental objects in the Tamo-Barg constructions of Locally Recoverable Codes (LRC). In this paper we classify all good polynomials up to degree $5$, providing explicit bounds on the maximal number $\ell$ of sets of size $r+1$ where a polynomial of degree $r+1$ is constant, up to $r=4$. This directly provides an explicit estimate (up to an error term of $O(\sqrt{q})$, with explict constant) for the maximal length and dimension of a Tamo-Barg LRC. Moreover, we explain how to construct good polynomials achieving these bounds. Finally, we provide computational examples to show how close our estimates are to the actual values of $\ell$, and we explain how to obtain the best possible good polynomials in degree $5$.