Maximally Recoverable Local Reconstruction Codes (LRCs) are codes designed for distributed storage to provide maximum resilience to failures for a given amount of storage redundancy and locality. An $(n,r,h,a,g)$-MR LRC has $n$ coordinates divided into $g$ local groups of size $r=n/g$, where each local group has `$a$' local parity checks and there are an additional `$h$' global parity checks. Such a code can correct `$a$' erasures in each local group and any $h$ additional erasures. Constructions of MR LRCs over small fields is desirable since field size determines the encoding and decoding efficiency in practice. In this work, we give a new construction of $(n,r,h,a,g)$-MR-LRCs over fields of size $q=O(n)^{h+(g-1)a-\lceil h/g\rceil}$ which generalizes a construction of Hu and Yekhanin (ISIT 2016). This improves upon state of the art when there are a small number of local groups, which is true in practical deployments of MR LRCs.

翻译：最大可恢复局部重建码（LRCs）是针对分布式存储设计的编码，旨在给定存储冗余和局部性的条件下提供最大故障恢复能力。一个$(n,r,h,a,g)$-MR LRC具有$n$个坐标，分为$g$个大小为$r=n/g$的局部组，其中每个局部组包含`$a$`个局部校验，并额外包含`$h$`个全局校验。此类编码能纠正每个局部组中的`$a$`个擦除以及任意$h$个额外擦除。在小域上构建MR LRCs具有实际意义，因为域大小直接影响编解码效率。本文提出了一种在域大小为$q=O(n)^{h+(g-1)a-\lceil h/g\rceil}$上的新型$(n,r,h,a,g)$-MR-LRC构造方法，该构造推广了Hu与Yekhanin（ISIT 2016）的工作。当局部组数量较少时，本构造优于现有技术，而这一条件在MR LRC的实际部署中普遍成立。