  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 $5$.
