We investigate the communication range of the nodes necessary for network connectivity, which we call bidirectional connectivity, in a simple setting. Unlike in most of existing studies, however, the locations or mobilities of the nodes may be correlated through group mobility: nodes are broken into groups, with each group comprising the same number of nodes, and lie on a unit circle. The locations of the nodes in the same group are not mutually independent, but are instead conditionally independent given the location of the group. We examine the distribution of the smallest communication range needed for bidirectional connectivity, called the critical transmission range (CTR), when both the number of groups and the number of nodes in a group are large. We first demonstrate that the CTR exhibits a parametric sensitivity with respect to the space each group occupies on the unit circle. Then, we offer an explanation for the observed sensitivity by identifying what is known as a very strong threshold and asymptotic bounds for CTR.

We study routing overhead due to location information collection and retrieval in mobile ad-hoc networks employing geographic routing with no hierarchy. We first provide a new framework for quantifying overhead due to control messages generated to exchange location information. Second, we compute the minimum number of bits required on average to describe the locations of a node, borrowing tools from information theory. This result is then used to demonstrate that the expected overhead is Ω (n1.5log (n)), where n is the number of nodes, under both proactive and reactive geographic routing, with the assumptions that 1) nodes' mobility is independent, and 2) nodes adjust their transmission range to maintain network connectivity. Finally, we prove that the minimum expected overhead under the same assumptions is Θ (n log (n)).

For an odd prime p, n=4k, and d=((p2k+1)/2)2, there are (p2k+1)/2 distinct decimated sequences s(dt+l), 0lesl2k+)/2,of a p-ary m-sequence s(t) of period pn-1 because gcd(d,pn-1)=(p2k+1)/2. In this paper, it is shown that the cross-correlation function between s(t) and s(dt+l),0lesl2k+1)/2, takes the values in {-1, -1 -radic(pn),-1+radic(pn),-1+2radic(pn)} and their cross-correlation distribution is also derived.

For an odd prime p, an even integer n, and d = pk+ 1 with gcd(n, k) = 1, there are p + 1 distinct decimated sequences s(dt + l), 0 les In- 1 since gcd(d, pn-1) = p + 1. In this paper, the cross-correlation distribution between a p-ary m-sequence s(t) and its p+1 distinct decimated sequences s(dt+l) is derived. The maximum magnitude of their cross-correlation values is l+p radic pnif I = 0 mod p + 1 for n = 0 mod 4 or I = (p + l)/2 mod p + 1 for n = 2 mod 4 and otherwise, 1 + radicpn. Also by using s(t) and s(dt + I), a new family of p-ary sequences of period pn-1 is constructed, whose family size is pnmiddot and Cmaxis 1+ pradicpn.

For an odd prime p, n=4k, and d=((p2k+1)/2)2, there are (p2k+1)/2distinct decimated sequences s(dt+l), 0lesl2k+1)/2, of a p-ary m-sequence s(t) of period pn-1 because gcd(d,pn-1)=(p2k+1)/2. In this paper, it is shown that the cross-correlation function between s(t) and s(dt+I), 0lesl2k+1)/2, takes the values in {-1, -1 - radicpn, -1+radicpn-1+2radicpn} and their cross-correlation distribution is also derived.