Spherical two-distance sets and related topics in harmonic analysis

This dissertation is devoted to the study of applications of harmonic analysis. The maximum size of spherical few-distance sets had been studied by Delsarte at al. in the 1970s. In particular, the maximum size of spherical two-distance sets in $\mathbb{R}^n$ had been known for $n \leq 39$ except $n=23$ by linear programming methods in 2008. Our contribution is to extend the known results of the maximum size of spherical two-distance sets in $\mathbb{R}^n$ when $n=23$, $40 \leq n \leq 93$ and $n \neq 46, 78$. The maximum size of equiangular lines in $\mathbb{R}^n$ had been known for all $n \leq 23$ except $n=14, 16, 17, 18, 19$ and $20$ since 1973. We use the semidefinite programming method to find the maximum size for equiangular line sets in $\mathbb{R}^n$ when $24 \leq n \leq 41$ and $n=43$. We suggest a method of constructing spherical two-distance sets that also form tight frames. We derive new structural properties of the Gram matrix of a two-distance set that also forms a tight frame for $\mathbb{R}^n$. One of the main results in this part is a new correspondence between two-distance tight frames and certain strongly regular graphs. This allows us to use spectral properties of strongly regular graphs to construct two-distance tight frames. Several new examples are obtained using this characterization. Bannai, Okuda, and Tagami proved that a tight spherical designs of harmonic index 4 exists if and only if there exists an equiangular line set with the angle $\arccos (1/(2k-1))$ in the Euclidean space of dimension $3(2k-1)^2-4$ for each integer $k \geq 2$. We show nonexistence of tight spherical designs of harmonic index $4$ on $S^{n-1}$ with $n\geq 3$ by a modification of the semidefinite programming method. We also derive new relative bounds for equiangular line sets. These new relative bounds are usually tighter than previous relative bounds by Lemmens and Seidel

New bounds for equiangular lines

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in $\mathbb{R}^n$, using semidefinite programming to improve the upper bounds on this quantity. Improvements are obtained in dimensions $24 \leq n \leq 136$. In particular, we show that the maximum number of equiangular lines in $\mathbb{R}^n$ is $276$ for all $24 \leq n \leq 41$ and is 344 for $n=43.$ This provides a partial resolution of the conjecture set forth by Lemmens and Seidel (1973).Comment: Minor corrections; added one new reference. To appear in "Discrete Geometry and Algebraic Combinatorics," A. Barg and O. R. Musin, Editors, Providence: RI, AMS (2014). AMS Contemporary Mathematics serie

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

Grassmannian Frames with Applications to Coding and Communication

For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with uniform tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on uniform tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to uniform tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.Comment: Submitted in June 2002 to Appl. Comp. Harm. Ana