Unbounded Localities and Unbounded Minimum Distances

A code over a finite field is called locally recoverable code (LRC) if every coordinate symbol can be determined by a small number (at most r, this parameter is called locality) of other coordinate symbols. For a linear 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 no optimal LRC code over a general finite field Fq with the length n ∼ q, the locality r ≥ 24 and the minimum distance d ≥ 7 has been constructed. In this paper for any given finite field Fq, any given r ∈ {1, 2, . . . , q− 1} and given d satisfying 3 ≤ d ≤ min{r + 1, q + 1 − r}, we give an optimal LRC code with length n = q(r + 1), locality r and minimum distance d. This is the only known family of optimal LRC codes with lengths n ∼ q and unbounded localities and minimum distances d ≥ 9. Hao Chen, Jian Weng and Weiqi Luo are with the College of Information Science and Technology/Cyber Security, Jinan University, Guangzhou, Guangdong Province 510632, China, haochen@jnu.edu.cn, cryptjweng@gmail.com, lwq@jnu.edu.cn. The research of Hao Chen was supported by NSFC Grants 11531002. The research of Jian Weng was supported by NSFC Distinguished Young Scholar Grant 61825203. The research of Weiqi Luo was supported by NSFC Grant 61877029.