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 $\delta$.

翻译：局部修复码（LRC）是一类广泛应用于分布式存储系统中的纠删码，能够在节点故障或数据丢失时高效恢复数据。2014年，Tamo和Barg基于多项式求值引入了类似里德-所罗门（RS-like）的Singleton最优$(r,\delta)$-LRC。这些构造依赖于所谓好多项式的存在性，该多项式在$\mathbb{F}_q$的若干两两不相交子集上取常数值。本文推广了上述RS-like LRC的构造，提出了能够获得更大码长的$(r,\delta)$-LRC新构造方法。这些新$(r,\delta)$-LRC均为距离最优的，即它们达到了本文将建立的最小距离上界。由于附加条件，该上界在某些情况下比Singleton型界更严格，且在某些情况下与Singleton型界一致。将这些构造与已知的特定形式显式好多项式相结合，我们能够获得多种具有新参数的显式Singleton最优$(r,\delta)$-LRC，其码长均大于Tamo和Barg提出的RS-like $(r,\delta)$-LRC的码长。注意经典RS码和RS-like LRC的码长均受限于域的大小。我们显式构造了码长$n=q-1+\delta$的Singleton最优$(r,\delta)$-LRC，其中$r,\delta\geq 2$为任意正整数且$(r+\delta-1)\mid (q-1)$。当$\delta$与$q$成比例时，该码长渐近地长于通过椭圆曲线构造的码长（后者至多为$q+2\sqrt{q}$），且对$r$和$\delta$的取值具有更大的灵活性。