发表机构
Indian Institute of Technology Roorkee（印度理工学院罗尔基分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Plotkin型构造所得二元码的恢复集结构，通过恢复超图建立服务速率区域界限，并递归推导Reed-Muller码的服务速率。
AI 中文摘要
在分布式存储系统中，冗余使得单个数据对象可以通过多个不相交的服务器组进行重构。由此产生的服务速率区域（SRR）刻画了系统能够同时支持而不使任何单个服务器过载的对象请求速率的所有组合。我们分析了通过Plotkin型构造获得的二元码的SRR。我们证明，对于Plotkin型码$\mathcal{C}_p$的信息对象，每个恢复集都诱导出由底层码$\mathcal{C}$所表征的同一数据对象的对应恢复集。此外，通过用$\mathcal{C}$的恢复集结构来刻画$\mathcal{C}_p$的恢复集结构，我们识别了该构造引入的额外恢复集。利用关联的恢复超图，我们建立了$\mathcal{C}_p$和迭代码$\mathcal{C}_{p^m}$的SRR相对于$\mathcal{C}$的SRR的界限。我们考虑一阶二元Reed-Muller码族，并展示如何通过我们对连续Plotkin型构造的结果，从R$(1,2)$的生成矩阵递归推导出任意$m$的R$(1,m)$的恢复结构，从而推导出其服务速率。
英文摘要
In distributed storage systems, redundancy enables a single data object to be reconstructed using several disjoint groups of servers. The resulting service rate region (SRR) captures all combinations of object request rates that the system can support simultaneously without overloading any individual server. We analyze the SRR of binary codes obtained via the Plotkin-type construction. We demonstrate that each recovery set for an information object of the Plotkin-type code $\mathcal{C}_p$ induces a corresponding recovery set for the same data object characterized by the underlying code $\mathcal{C}$. Furthermore, by characterizing the recovery sets structure of $\mathcal{C}_p$ in terms of the recovery set structure of $\mathcal{C}$, we identify the additional recovery sets introduced by this construction. Using the associated recovery hypergraphs, we establish bounds on the SRRs of $\mathcal{C}_p$ and iterated code $\mathcal{C}_{p^m}$ in terms of the SRR of $\mathcal{C}$. We consider the family of first-order binary Reed-Muller codes and show how the recovery structure, and consequently, the service rates of R$(1,m)$ for arbitrary $m$ can be recursively derived from a generator matrix of R$(1,2)$ through our results for successive Plotkin-type constructions.
