A Gray Code for the Shelling Types of the Boundary of a Hypercube

We consider two shellings of the boundary of the hypercube equivalent if one can be transformed into the other by an isometry of the cube. We observe that a class of indecomposable permutations, bijectively equivalent to standard double occurrence words, may be used to encode one representative from each equivalence class of the shellings of the boundary of the hypercube. These permutations thus encode the shelling types of the boundary of the hypercube. We construct an adjacent transposition Gray code for this class of permutations. Our result is a signed variant of King's result showing that there is a transposition Gray code for indecomposable permutations

Weighing matrices and spherical codes

Mutually unbiased weighing matrices (MUWM) are closely related to an antipodal spherical code with 4 angles. In the present paper, we clarify the relationship between MUWM and the spherical sets, and give the complete solution about the maximum size of a set of MUWM of weight 4 for any order. Moreover we describe some natural generalization of a set of MUWM from the viewpoint of spherical codes, and determine several maximum sizes of the generalized sets. They include an affirmative answer of the problem of Best, Kharaghani, and Ramp.Comment: Title is changed from "Association schemes related to weighing matrices

Experimental study of energy-minimizing point configurations on spheres

In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic

Quantum Random Access Codes with Shared Randomness

We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79) QRACs match this upper bound). We discuss some particular constructions for several small values of n. We also study the classical counterpart of this model where n bits are encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal construction for such codes and find their success probability exactly--it is less than in the quantum case. Interactive 3D quantum random access codes are available on-line at http://home.lanet.lv/~sd20008/racs .Comment: 51 pages, 33 figures. New sections added: 1.2, 3.5, 3.8.2, 5.4 (paper appears to be shorter because of smaller margins). Submitted as M.Math thesis at University of Waterloo by M

Density of Spherically-Embedded Stiefel and Grassmann Codes

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE Transactions on Information Theor

Structural stability of meandering-hyperbolic group actions

In his 1985 paper Sullivan sketched a proof of his structural stability theorem for group actions satisfying certain expansion-hyperbolicity axioms. In this paper we relax Sullivan's axioms and introduce a notion of meandering hyperbolicity for group actions on general metric spaces. This generalization is substantial enough to encompass actions of certain non-hyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.Comment: 58 pages, 5 figures; [v2] Corollary 3.19 is wrong and thus removed; [v3] Introduced a new notion of meandering hyperbolicity, generalized the main structural stability theorem even further, and added a new Section 5 on uniform lattices and their structural stabilit

Improved Orientation Sampling for Indexing Diffraction Patterns of Polycrystalline Materials

Orientation mapping is a widely used technique for revealing the microstructure of a polycrystalline sample. The crystalline orientation at each point in the sample is determined by analysis of the diffraction pattern, a process known as pattern indexing. A recent development in pattern indexing is the use of a brute-force approach, whereby diffraction patterns are simulated for a large number of crystalline orientations, and compared against the experimentally observed diffraction pattern in order to determine the most likely orientation. Whilst this method can robust identify orientations in the presence of noise, it has very high computational requirements. In this article, the computational burden is reduced by developing a method for nearly-optimal sampling of orientations. By using the quaternion representation of orientations, it is shown that the optimal sampling problem is equivalent to that of optimally distributing points on a four-dimensional sphere. In doing so, the number of orientation samples needed to achieve a indexing desired accuracy is significantly reduced. Orientation sets at a range of sizes are generated in this way for all Laue groups, and are made available online for easy use.Comment: 11 pages, 7 figure