AI 中文总结
本文提出两种构造最优访问协作MSR码的方法，通过奇偶校验矩阵设计和统一变换框架，显著降低子包化水平，并涵盖现有结构作为特例。
AI 中文摘要
协作MSR码是一种存储码，能够以协作方式对任意$h\geq2$个节点擦除实现最优带宽修复，同时保持作为$[n,k]$ MDS码的最小存储。每个码坐标（节点）假定存储一个由$\ell$个符号组成的数组，其中$\ell$被称为子包化（sub-packetization）。为了解决磁盘IO（输入/输出）能力问题，如果在节点修复过程中每个辅助节点访问的数据量达到该数量的下界，则称协作MSR码具有最优访问性质。本文聚焦于降低最优访问协作MSR码的子包化水平。我们通过两种方法提出了最优访问协作MSR码的新构造。首先，我们通过设计其奇偶校验矩阵提出一种直接显式构造。这种奇偶校验矩阵通过重复使用两个关键的奇偶校验矩阵作为构建块来构建。其次，我们提出一个通用变换框架。从任意$[n+d-k,d]$ MDS标量码出发，通过系统应用两个基本变换，可以推导出最终的协作MSR码。两种方法均产生$(n,k,\ell=\delta^m)$最优访问协作MSR码，其中$\delta=d-k+h$且$m=\binom{n}{h}-\lfloor\frac{n}{\delta}\rfloor(\binom{\delta}{h}-1)$。与现有技术（$\ell=\delta^{\binom{n}{h}}$）相比，推导出的码可以将子包化$\ell$减少$1/\delta^{\lfloor\frac{n}{\delta}\rfloor(\binom{\delta}{h}-1)}$的比例，其中$\delta=d-k+h$。此外，我们还表明，一些先前的最优访问协作MSR码和$h=1$的最优访问MSR码结构是作为我们变换构造的特例包含在内的。最后，我们注意到所有构造都建立在大小$\geq n+d-k$的有限域上。
英文摘要
Cooperative MSR codes are a kind of storage codes which enable optimal-bandwidth repair of any $h\geq2$ node erasures in a cooperative way, while retaining the minimum storage as an $[n,k]$ MDS code. Each code coordinate (node) is assumed to store an array of $\ell$ symbols, where $\ell$ is termed as sub-packetization. To address the disk IO (input/output) capability, a cooperative MSR code is said to have optimal-access property, if during node repair, the amount of data accessed at each helper node meets a lower bound on this quantity. In this paper, we focus on reducing the sub-packetization level of optimal-access cooperative MSR codes. We propose new constructions of optimal-access cooperative MSR codes through two methods. At first, we propose a direct explicit construction by designing its parity-check matrix. Such parity-check matrix is built by repeatedly employing two crucial parity-check matrices as building blocks. Secondly, we propose a generic transformation framework. Starting from an arbitrary $[n+d-k,d]$ MDS scalar code, one can derive a final cooperative MSR code by systematically applying two basic transformations. Both approaches yield $(n,k,\ell=δ^m)$ optimal-access cooperative MSR codes with $δ=d-k+h$ and $m=\binom{n}{h}-\lfloor\frac{n}δ\rfloor(\binomδ{h}-1)$. Compared with the state of the art (with $\ell=δ^{\binom{n}{h}}$), the derived codes can reduce the sub-packetization $\ell$ by a fraction of $1/δ^{\lfloor\frac{n}δ\rfloor(\binomδ{h}-1)}$, where $δ=d-k+h$. Moreover, we also show that some previous code structures of optimal-access cooperative MSR codes and optimal-access MSR codes with $h=1$ are included as special cases of our transformation construction. At last, we note that all of the constructions are built over a finite field of linear size $\geq n+d-k$.
