Abstract
Asynchronous channel hopping (ACH) systems are widely used for the blind rendezvous among secondary users in cognitive radio networks without requirement for global synchronization and common control channels. We can view an ACH system with n channels as a set of sequences of a common period t on an alphabet of size n satisfying certain rotation closure properties. For any two distinct sequences u, v in an ACH system \(\mathcal {S}\), every l ∈{0,1,⋯ , t − 1} and any letter j in the alphabet, if there always exists i such that the i-th entries of u and Ll(v) are identical to j where Ll(v) denotes the cyclic shift of v by l, then we say that \(\mathcal {S}\) is a complete ACH system. Such a system can guarantee rendezvous between any two secondary users who share at least one common channel. It is well known that \(t\geqslant n^{2}\). When the equality holds, \(\mathcal {S}\) is called a perfect ACH system. By applying characters of cyclic groups, we obtain a new upper bound on the number of sequences in a perfect ACH system. Furthermore, when q is a prime power and n = q − 1, we present a construction of homogeneous complete ACH systems of period t = 2n2 + o(n2).
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Acknowledgments
This work is partially supported by the National Key R&D Program of China (No. 2017YFB0802000) and the National Natural Science Foundation of China (No. 11771451). Yue Zhou is supported by the Alexander von Humboldt Foundation.
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This article is part of the Topical Collection on Special Issue on Sequences and Their Applications
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Gao, Z., Li, C. & Zhou, Y. Upper bounds and constructions of complete Asynchronous channel hopping systems. Cryptogr. Commun. 11, 299–312 (2019). https://doi.org/10.1007/s12095-018-0295-4
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DOI: https://doi.org/10.1007/s12095-018-0295-4