Abstract

Fractional repetition codes (FRCs) are a special family of storage codes with the repair-by-transfer property in distributed storage systems. Constructions of FRCs are naturally related to combinatorial designs, graphs, and hypergraphs. In this paper, we consider an extremal problem on regular graphs related to FRCs where each packet is stored on \(\rho =2\) nodes. The problem asks for the minimum number of vertices in an \(\alpha \) -regular graph such that any k vertices induce at most \(\delta \) edges, where \(\alpha \) , k, and \(\delta \) are given. Such a problem is closely related to (and can be seen as a generalization of) the classical cage problem, and its solution indicates the minimum number of nodes in an FRC-based distributed storage system. In addition, we further consider FRCs with \(\rho \ge 3\) and generalize the extremal problem to a linear hypergraph version.

中文翻译：

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与分数重复码相关的极值正则图和超图

摘要

分数重复码（FRC）是分布式存储系统中一类特殊的存储码，具有传输修复特性。FRC 的构造自然与组合设计、图和超图相关。在本文中，我们考虑与 FRC 相关的正则图上的极值问题，其中每个数据包都存储在\(\rho =2\)节点上。该问题要求\(\alpha \)正则图中的最小顶点数，使得任何k 个顶点最多产生\(\delta \)条边，其中\(\alpha \)、k和\(\给出了delta \) 。该问题与经典的笼问题密切相关（并且可以看作是其推广），其解表明基于FRC的分布式存储系统中的最小节点数。此外，我们进一步考虑具有\(\rho \ge 3\)的 FRC ，并将极值问题推广到线性超图版本。