Robert B. Ellis

Random Geometric Graph Diameter in the Unit Ball

The unit ball random geometric graph

Asymmetric binary covering codes

G=G^d_p(\lambda,n)

Ulam's pathological liar game with one half-lie

has as its vertices n points distributed independently and uniformly in the unit ball in

The Rényi-Ulam pathological liar game with a fixed number of lies

{\Bbb R}^d

Monitoring schedules for randomly deployed sensor networks

, with two vertices adjacent if and only if their ℓp-distance is at most λ. Like its cousin the Erdos-Renyi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone we determine upper and lower bounds for the graph diameter of G. Specifically, almost always,

Random geometric graph diameter in the unit disk with l p metric

{\rm diam}_p({\bf B})(1-o(1))/\lambda\leq {\rm diam}(G) \leq {\rm diam}_p({\bf B})(1+O((\ln \ln n/{\rm ln}\,n)^{1/d}))/\lambda

Multiuser network coding based on DFT matrices design

, where

Embedded space–time codes for multiple-input–multiple-output systems in Rayleigh fading

{\rm diam}_p({\bf B})

Space time block code for four time slots and two transmit antennas

is the ℓp-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.

The structure of super line graphs

An asymmetric binary covering code of length n and radius R is a subset C of the n-cube Qn such that every vector x ∈ Qn can be obtained from some vector c ∈ c by changing at most R 1s of c to 0s, where R is as small as possible. K+ (n, R) is defined as the smallest size of such a code. We show K+(n, R) ∈ Θ (2n/nR) for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K+(n,n -R) = R+ 1 for constant coradius R iff n ≥ R(R + 1)/2. These two results are extendetd to near-constant R and R, respectively. Various bounds on K+ are given in terms of the total number of 0s or 1s in a minimal code. The dimension of a minimal asymmetric linear binary code ([n, R]+-code) is determined to be min{0, n - R}. We conclude by discussing open problems and techniques to compute explicit values for K+, giving a table of best-known bounds.