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Asynchronous channel hopping systems from difference sets

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Abstract

In cognitive radio networks, the process for any two unlicensed (secondary) users (SU) to establish links through a common channel is called a rendezvous. It is hard to employ a common control channel to solve the rendezvous problem, because of the bottleneck of the single control channel. Asynchronous channels hopping (ACH) systems have been proposed and investigated to guarantee rendezvous without requirement of global synchronization and common control channels. An ACH system with N channels can be mathematically interpreted as a set of sequences of the same period T on the alphabet \(\{0,1,\ldots , N-1\}\) satisfying certain rotation closure properties. For each \(l\in \{0,1,\ldots , T-1 \}\), each \(j\in \{0,1,\ldots , N-1 \}\) and any two distinct sequences \(\mathbf u\), \(\mathbf v\) in an ACH system H, if there always exists i such that the i-th entries of \(\mathbf u\) and \(L^{l}(\mathbf v)\) are both identical to j where \(L^{l}(\mathbf v)\) denotes the cyclic shift of \(\mathbf v\) by l, then H is called a complete ACH system, which guarantees rendezvous between any two SUs who share at least one common channel. In this paper, we prove some properties of such systems. Moreover, we investigate the complete ACH systems, in each of which all SUs repeat the same (global) sequence associated with the system. One challenging research problem is to construct such ACH systems of period \(T=\beta N^2 + o(N^2)\) such that \(\beta \) is as small as possible. By applying mathematical tools such as group rings and (relative, relaxed) difference sets from combinatorial design theory, we obtain two constructions of them in which \(\beta =1,2\) respectively. Finally, we also present a simple and powerful approach to combining known ACH systems to produce new ones.

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Acknowledgments

We would like to thank the anonymous referees for their valuable comments and suggestions on the manuscript. Xiaotian Chen is partially supported by the National Natural Science Foundation of China (No. 61501476). Yue Zhou is partially supported by the National Natural Science Foundation of China (Nos. 11401579, 11531002) and the Research Project of MIUR (Italian Office for University and Research) “Strutture geometriche, Combinatoria e loro Applicazioni” 2012.

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Authors and Affiliations

  1. College of Electronic Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    Xiaotian Chen

  2. College of Science, National University of Defense Technology, Changsha, 410072, China

    Yue Zhou

  3. Dipartimento di Mathematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, 80126, Napoli, Italy

    Yue Zhou

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  1. Xiaotian Chen
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  2. Yue Zhou
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Correspondence to Yue Zhou.

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Communicated by J. D. Key.

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Chen, X., Zhou, Y. Asynchronous channel hopping systems from difference sets. Des. Codes Cryptogr. 83, 179–196 (2017). https://doi.org/10.1007/s10623-016-0221-8

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