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Extremal regular graphs and hypergraphs related to fractional repetition codes

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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.

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Notes

  1. Note that if p is adjacent to exactly four vertices among the \(v_i\)-parts, then we can further deduce that the adjacency of p can only be of three forms: adjacent to two vertices from the \(v_1\)-part and two vertices from the \(v_3\)-part, adjacent to two vertices from the \(v_2\)-part and two vertices from the \(v_4\)-part, or adjacent with one vertex from each of the four parts. This argument will also be use in the next two corollaries.

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Authors and Affiliations

  1. Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education, School of Cyber Science and Technology, Shandong University, Qingdao, 266237, Shandong, China

    Hongna Yang, Yiwei Wang & Yiwei Zhang

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  1. Hongna Yang
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  2. Yiwei Wang
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Correspondence to Yiwei Zhang.

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Communicated by G. Ge.

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Research supported in part by the National Key Research and Development Program of China under Grant 2022YFA1004900 and Grant 2021YFA1001000, in part by the National Natural Science Foundation of China under Grant 12231014 and Grant 12001323, and in part by the Shandong Provincial Natural Science Foundation under Grant No. ZR2021YQ46. Part of this work [25] was presented at IEEE International Symposium on Information Theory (ISIT), 2022.

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Yang, H., Wang, Y. & Zhang, Y. Extremal regular graphs and hypergraphs related to fractional repetition codes. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01370-5

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