Good polynomials for optimal LRC of low locality

Abstract

According to a magnific method due to I. Tamo and A. Barg, a class of polynomials over finite fields, called good polynomials, was introduced and used to construct optimal Locally Recoverable Codes (LRC), which have been developed and exploited in distributed storage. An important derived algebraic problem is, for a given finite field $$\mathbb {F}_q$$ and a fixed integer r, to find a polynomial of degree $$r+1$$ that is constant on as many subsets of $$\mathbb {F}_q$$ as possible of size $$r+1$$ . Compared to the literature on this topic, our main contribution is introducing a new parameter that measures how “good” a polynomial is in the sense of LRC. Our new approach allows us to characterize completely good polynomials of a low degree over finite fields and, next, to derive new constructions of such polynomials, leading to optimal LRC with new flexible localities. Specifically, several good polynomials of degree at most 6 are studied and described precisely in this paper.