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Good polynomials are the fundamental objects in the Tamo-Barg constructionsof Locally Recoverable Codes (LRC). In this paper we classify all goodpolynomials 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, upto $r=4$. 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 aTamo-Barg LRC. Moreover, we explain how to construct good polynomials achievingthese bounds. Finally, we provide computational examples to show how close ourestimates are to the actual values of $\ell$, and we explain how to obtain thebest possible good polynomials in degree $5$.

Optimal Selection for Good Polynomials of Degree up to Five