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

Biangular vectors

viii, 133 leaves ; 29 cmThis thesis introduces unit weighing matrices, a generalization of Hadamard matrices. When dealing with unit weighing matrices, a lot of the structure that is held by Hadamard matrices is lost, but this loss of rigidity allows these matrices to be used in the construction of certain combinatorial objects. We are able to fully classify these matrices for many small values by defining equivalence classes analogous to those found with Hadamard matrices. We then proceed to introduce an extension to mutually unbiased bases, called mutually unbiased weighing matrices, by allowing for different subsets of vectors to be orthogonal. The bounds on the size of these sets of matrices, both lower and upper, are examined. In many situations, we are able to show that these bounds are sharp. Finally, we show how these sets of matrices can be used to generate combinatorial objects such as strongly regular graphs and association schemes

Infinite families of optimal systems of biangular lines related to representations of SL(2,Fq)\textrm{SL}(2,\mathbb{F}_q)

A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds for coherence. In this paper two two infinite families of real and complex biangular line packings are presented. New packings achieve equality in the real or complex second Levenshtein bound respectively. Both infinite families are constructed by analyzing well known representations of the finite groups $\textrm{SL}(2,\mathbb{F}_q)$. Until now the only known infinite familes meeting the second Levenshtein bounds were related to the maximal sets of mutually unbiased bases (MUB). Similarly to the line packings related to the maximal sets of MUBs, the line packings presented here are related to the maximal sets of mutually unbiased weighing matrices. Another similarity is that the new packings are projective 2-designs. The latter property together with sufficiently large cardinalities of the new packings implies some improvement on largest known cardinalities of real and complex biangular tight frames (BTF)