Energy bounds for codes and designs in Hamming spaces

We obtain universal bounds on the energy of codes and for designs in Hamming spaces. Our bounds hold for a large class of potential functions, allow unified treatment, and can be viewed as a generalization of the Levenshtein bounds for maximal codes.Comment: 25 page

Equiangular Lines and Spherical Codes in Euclidean Space

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $\theta$. Moreover, this is achievable only for $\theta = \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k)$ lines in $\mathbb{R}^n$ having these angles. This bound, conjectured by Bukh, substantially improves the estimate of Delsarte, Goethals and Seidel from 1975. Various extensions of these results to the more general setting of spherical codes will be discussed as well.Comment: 24 pages, 0 figure

Spherical sets avoiding a prescribed set of angles

Let $X$ be any subset of the interval $[-1,1]$. A subset $I$ of the unit sphere in $R^n$ will be called \emph{$X$-avoiding} if $\notin X$ for any $u,v \in I$. The problem of determining the maximum surface measure of a $\{ 0 \}$-avoiding set was first stated in a 1974 note by Witsenhausen; there the upper bound of $1/n$ times the surface measure of the sphere is derived from a simple averaging argument. A consequence of the Frankl-Wilson theorem is that this fraction decreases exponentially, but until now the $1/3$ upper bound for the case $n=3$ has not moved. We improve this bound to $0.313$ using an approach inspired by Delsarte's linear programming bounds for codes, combined with some combinatorial reasoning. In the second part of the paper, we use harmonic analysis to show that for $n\geq 3$ there always exists an $X$-avoiding set of maximum measure. We also show with an example that a maximiser need not exist when $n=2$.Comment: 21 pages, 3 figure

Some constructions of superimposed codes in Euclidean spaces

AbstractWe describe three new methods for obtaining superimposed codes in Euclidean spaces. With help of them we construct codes with parameters improving upon known constructions. We also prove that the spherical simplex code is not optimal as superimposed code at least for dimensions greater than 9

Einstein equations in the null quasi-spherical gauge III: numerical algorithms

We describe numerical techniques used in the construction of our 4th order evolution for the full Einstein equations, and assess the accuracy of representative solutions. The code is based on a null gauge with a quasi-spherical radial coordinate, and simulates the interaction of a single black hole with gravitational radiation. Techniques used include spherical harmonic representations, convolution spline interpolation and filtering, and an RK4 "method of lines" evolution. For sample initial data of "intermediate" size (gravitational field with 19% of the black hole mass), the code is accurate to 1 part in 10^5, until null time z=55 when the coordinate condition breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed