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 -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 . In this paper, we extend the aforementioned constructions of RS-like LRCs and propose new constructions of -LRCs whose code length can be larger. These new -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. Combining our constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal -LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like -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 -LRCs with length for any positive integers and . We also show the existence of Singleton-optimal -LRCs with length over () provided , and . When is proportional to , they are asymptotically longer than that constructed via elliptic curves whose length is at most . Besides, they allow more flexibility on the values of and .

中文翻译：

---

通过良好多项式构建最优 (r,δ)-LRC

本地可修复码（LRC）是一类广泛应用于分布式存储系统的纠删码，可以在节点故障或数据丢失的情况下有效恢复数据。2014 年，Tamo 和 Barg 引入了基于多项式评估的 Reed-Solomon-like (RS-like) Singleton-optimal-LRC。这些构造依赖于所谓的良好多项式的存在，该多项式对于 的一些成对不相交子集中的每一个都是恒定的。在本文中，我们扩展了上述类RS LRC的结构，并提出了码长可以更大的新的-LRC结构。这些新的-LRC都是距离最优的，即它们达到了本文将建立的最小距离的上限。由于额外的条件，这个界限在某些情况下比 Singleton 类型的界限更尖锐，在某些情况下它与 Singleton 类型的界限一致。将我们的构造与已知的特殊形式的显式好多项式相结合，我们可以得到各种具有新参数的显式单例最优-LRC，其代码长度都大于Tamo和Barg引入的类RS-LRC构造的代码长度。请注意，经典 RS 代码和类似 RS 的 LRC 的代码长度均受字段大小的限制。我们显式地构造单例最优-LRC，其长度对于任何正整数 和 。我们还证明了单例最优-LRC 的存在，其长度超过 () 提供的 、 和 。当 与 成比例时，它们渐近地长于通过长度至多为 的椭圆曲线构造的曲线。此外，它们允许 和 的值具有更大的灵活性。