New constructions of optimal $(r,δ)$-LRCs via good polynomials

Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal $(r,\delta)$-LRCs based on polynomial evaluation. These constructions rely on the existence of so-called good polynomial that is constant on each of some pairwise disjoint subsets of $\mathbb{F}_q$. In this paper, we extend the aforementioned constructions of RS-like LRCs and proposed new constructions of $(r,\delta)$-LRCs whose code length can be larger. These new $(r,\delta)$-LRCs are all distance-optimal, namely, they attain an upper bound on the minimum distance, that will be established in this paper. This bound is sharper than the Singleton-type bound in some cases owing to the extra conditions, it coincides with the Singleton-type bound for certain cases. Combing these constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal $(r,\delta)$-LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like $(r,\delta)$-LRCs introduced by Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs are both bounded by the field size. We explicitly construct the Singleton-optimal $(r,\delta)$-LRCs with length $n=q-1+\delta$ for any positive integers $r,\delta\geq 2$ and $(r+\delta-1)\mid (q-1)$. When $\delta$ is proportional to $q$, it is asymptotically longer than that constructed via elliptic curves whose length is at most $q+2\sqrt{q}$. Besides, it allows more flexibility on the values of $r$ and $\delta$.