Long Optimal or Small-Defect LRC Codes with Unbounded Minimum Distances

Abstract

For a linear locally recoverable code (LRC) with length n, dimension k and locality r its minimum distance d satisfies d ≤ n−k+2−⌈ k r ⌉. A code attaining this bound is called optimal. Many families of optimal locally recoverable codes have been constructed by using different techniques in finite fields or algebraic curves. However in previous constructions of length n >> q optimal LRC codes minimum distances are only few constants smaller than 9. No optimal LRC code over a general finite field F q with the length n ∼ q 2 and the minimum distance d ≥ 9 has been constructed. In this paper we present a general construction of optimal LRC codes over arbitrary finite fields. Over any given finite field F q , for any given r ∈ {1, 2, . . . , q − 1} and given d satisfying 3 ≤ d ≤ min{r + 1, q + 1 − r}, we construct explicitly an optimal LRC code with length n = q(r + 1), locality r and minimum distance d. We also gives an asymptotic bound for q-ary r ≤ q − 1-locality LRC codes better than the previously konwn bound. Many long r-locality LRC codes with small defect s = n−k+2−⌈ k r ⌉−d are also constructed.