论文标题
最佳$（R，δ）$ - LRC的新结构通过良好的多项式
New constructions of optimal $(r,δ)$-LRCs via good polynomials
论文作者
论文摘要
本地维修代码（LRC）是一类擦除代码，这些代码广泛用于分布式存储系统中，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根据多项式评估引入了Reed-Solomon样（RS样）单顿式$（R，δ）$ -LRCS。这些构造依赖于所谓的良好多项式的存在，在$ \ mathbb {f} _q $的每个成对分离子集中都是恒定的。在本文中，我们扩展了上述类似RS的LRC的结构，并提出了$（r，δ）$ -LRC的新结构，其代码长度可以更大。这些新的$（R，δ）$ - LRC均为距离最佳，即它们在本文中将建立在最小距离上的上限。由于额外的条件，这种界限比单身型的型单型型型，它与某些情况下的单身型型相吻合。将这些结构与已知的特殊形式的已知良好多项式梳理，我们可以获得具有新参数的各种明确的单胎$（r，δ）$ -LRCS，其代码长度都大于tamo和tamo和barg和barg和barg和barg ofd of t like $（r，δ）$ -LRCS构建的代码长度。请注意，经典的RS代码和类似RS的LRC的代码长度均由字段大小界定。我们明确地构建了singleton-optimal $（r，δ）$ - lrcs，长度为$ n = q-1+δ$，对于任何正整数$ r，δ\ geq 2 $和$（r+Δ1）\ mid（q-1）$。当$δ$与$ q $成正比时，它的渐近性比通过椭圆曲线构建的长度最多为$ q+2 \ sqrt {q} $。此外，它允许在$ r $和$δ$的值上更灵活。
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 $(r,δ)$-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 $(r,δ)$-LRCs whose code length can be larger. These new $(r,δ)$-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 $(r,δ)$-LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like $(r,δ)$-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 $(r,δ)$-LRCs with length $n=q-1+δ$ for any positive integers $r,δ\geq 2$ and $(r+δ-1)\mid (q-1)$. When $δ$ 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 $δ$.
