Optimal Selection for Good Polynomials of Degree up to Five

04/03/2021

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by   Austin Dukes, et al.

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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 ℓ 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(√(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 ℓ, and we explain how to obtain the best possible good polynomials in degree 5.