Block-Transitive Designs in Affine Spaces

This paper deals with block-transitive $t$-$(v,k,\lambda)$ designs in affine spaces for large $t$, with a focus on the important index $\lambda=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs, except possibly when the group is one-dimensional affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.Comment: 10 pages; to appear in: "Designs, Codes and Cryptography

Restricted Isometries for Partial Random Circulant Matrices

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde

Gabor frames in finite dimensions

Gabor frames have been extensively studied in time-frequency analysis over the last 30 years. They are commonly used in science and engineering to synthesize signals from, or to decompose signals into, building blocks which are localized in time and frequency. This chapter contains a basic and self-contained introduction to Gabor frames on finite-dimensional complex vector spaces. In this setting, we give elementary proofs of the central results on Gabor frames in the greatest possible generality; that is, we consider Gabor frames corresponding to lattices in arbitrary finite Abelian groups. In the second half of this chapter, we review recent results on the geometry of Gabor systems in finite dimensions: the linear independence of subsets of its members, their mutual coherence, and the restricted isometry property for such systems. We apply these results to the recovery of sparse signals, and discuss open questions on the geometry of finite-dimensional Gabor systems