Optimal (<i>r</i>, δ)-Locally Repairable Codes from Reed-Solomon Codes

Abstract

For an [n, k, d] (r, δ)-locally repairable codes ((r, δ)-LRCs) , its minimum distance d satisfies the Singleton-like bound. The construction of optimal (r, δ)-LRC, attaining this Singleton-like bound, is an important research problem in recent years for thier applications in distributed storage systems. In this letter, we use Reed-Solomon codes to construct two classes of optimal (r, δ)-LRCs. The optimal LRCs are given by the evaluations of multiple polynomials of degree at most r - 1 at some points in 𝔽q. The first class gives the [(r + δ - 1)t, rt - s, δ + s] optimal (r, δ)-LRC over 𝔽q provided that r + δ + s - 1 ≤ q, s ≤ δ, s < r, and any positive t. The code length is unbounded. The second class gives the [r +r' + d + δ - 2, r + r', d] optimal (r, δ)-LRC over 𝔽q provided that r - r' ≥ d - δ and r + d - 1 ≤ q + 1, which will produce optimal (r, δ)-LRCs with large minimum distance.