Optimal Selection for Good Polynomials of Degree up to Five

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 $5$.