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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 1/2017

14-05-2016

Asynchronous channel hopping systems from difference sets

Authors: Xiaotian Chen, Yue Zhou

Published in: Designs, Codes and Cryptography | Issue 1/2017

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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.
previous article On the automorphism groups of the -linear 1-perfect and Preparata-like codes
next article Explicit factorizations of cyclotomic polynomials over finite fields
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Metadata
Title
Asynchronous channel hopping systems from difference sets
Authors
Xiaotian Chen
Yue Zhou
Publication date
14-05-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 1/2017
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-016-0221-8

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