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,…,N-1}\documentclass[12pt]{minimal}
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\begin{document}$$\{0,1,\ldots , N-1\}$$\end{document} satisfying certain rotation closure properties. For each l∈{0,1,…,T-1}\documentclass[12pt]{minimal}
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\begin{document}$$l\in \{0,1,\ldots , T-1 \}$$\end{document}, each j∈{0,1,…,N-1}\documentclass[12pt]{minimal}
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\begin{document}$$j\in \{0,1,\ldots , N-1 \}$$\end{document} and any two distinct sequences u\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf u$$\end{document}, v\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf v$$\end{document} in an ACH system H, if there always exists i such that the i-th entries of u\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf u$$\end{document} and Ll(v)\documentclass[12pt]{minimal}
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\begin{document}$$L^{l}(\mathbf v)$$\end{document} are both identical to j where Ll(v)\documentclass[12pt]{minimal}
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\begin{document}$$L^{l}(\mathbf v)$$\end{document} denotes the cyclic shift of v\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf v$$\end{document} 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=βN2+o(N2)\documentclass[12pt]{minimal}
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\begin{document}$$T=\beta N^2 + o(N^2)$$\end{document} such that β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} 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 β=1,2\documentclass[12pt]{minimal}
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\begin{document}$$\beta =1,2$$\end{document} respectively. Finally, we also present a simple and powerful approach to combining known ACH systems to produce new ones.
