Optimal selection for good polynomials of degree up to five

An $(r,\ell )$-good polynomial is a polynomial of degree $r+1$ that is constant on $\ell$ subsets of ${\mathbb{F}}_q$, each of size $r+1$. For any positive integer $r\le 4$ we provide an $(r,\ell )$-good polynomial such that $\ell =C_{r}q+O(\sqrt{q})$, with $C_r$ maximal. 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. Our results complete the study by Chen et al. (Des Codes Cryptogr 89(7):1639–1660, 2021), providing $(r,\ell )$-good polynomials of degree up to 5, with $\ell$ maximal (up to an error term of $\sqrt{q}$), and our methods are independent.