Optimal selection for good polynomials of degree up to five

An $$(r,\ell )$$ -good polynomial is a of degree $$r+1$$ that constant on $$\ell $$ subsets $$\mathbb F_q$$ , each size . For any positive integer $$r\le 4$$ we provide an such =C_rq+O(\sqrt{q})$$ with $$C_r$$ maximal. This directly provides explicit estimate (up to error term $$O(\sqrt{q})$$ explict constant) for the maximal length and dimension Tamo–Barg LRC. Moreover, explain how construct good polynomials achieving these bounds. Finally, computational examples show close our estimates are actual values obtain best possible in 5. Our results complete study by Chen et al. (Des Codes Cryptogr 89(7):1639–1660, 2021), providing up 5, $$\sqrt{q}$$ ), methods independent.