New Constructions of Optimal Locally Recoverable Codes via Good Polynomials

Abstract

In recent literature, a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality) is presented. The key ingredient for constructing such optimal linear LRC codes is the so-called r-good polynomials, where r is equal to the locality of the LRC code. However, given a prime p, known constructions of r-good polynomials over some extension field of Fp exist only for some special integers r, and the problem of constructing optimal LRC codes over small field for any given locality is still open. In this paper, by using function composition, we present two general methods of designing good polynomials, which lead to three new constructions of r-good polynomials. Such polynomials bring new constructions of optimal LRC codes. In particular, our constructed polynomials as well as the power functions yield optimal (n, k, r) LRC codes over Fq for all positive integers r as localities, where q is near the code length n.