Long Optimal and Small-Defect LRC Codes With Unbounded Minimum Distances

Abstract

For a linear locally recoverable (LRC) code 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 only optimal LRC codes with lengths n >> q and minimum distances restricted to few constants smaller than 9, have been given in previous constructions. No optimal LRC code over a general finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> with the length n ~ q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and the minimum distance d ≥ 9 has been constructed. In this article we present a general construction of optimal LRC codes over arbitrary finite fields. Over any given finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , 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 give an asymptotic bound of q-ary LRC codes with locality r (r ≤ q-1), which is better than some known previous asymptotic bounds in some parameter range. Moreover many long LRC codes with locality r (r ≤ q-1) and small defect s=n-k+2-⌈k/r⌉-d are also constructed from algebraic curves with many rational points.