On biunimodular vectors for unitary matrices

A biunimodular vector of a unitary matrix $A \in U(n)$ is a vector v \in \mathbb{T}^n\subset\bc^n such that $Av \in \mathbb{T}^n$ as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences. Here, we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices. This search can be motivated in various ways. The main motivation is provided by the fact, that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices

New weighing matrices

New weighing matrices and skew weighing matrices are given for many orders 4t ≤ 100. These are constructed by finding new sequences with zero autocorrelation. These results enable us to determine for the first time that for 4t ≤ 84 a W{4t,k) exists for all k = 1, ... ,4t -1 and also that there exists a skew-weighing matrix (also written as an OD(4t;1,k)) for 4t ≤ 80, t odd, k = a2 + b2 + c2,a,b,c integers except k = 4t - 2 must be the sum of two squares

Orthogonal designs in linear models and sequences with zero autocorrelation

In factorial experiments the use of orthogonal designs provides better estimates for the main effects and interactions. In 2k factorial experiments we obtain orthogonal designs from Hadamard matrices. Sequences with zero autocorrelation can be used to construct Hadamard matrices. In this paper new sequences with zero autocorrelation which are called [delta]-sequences are employed to construct some new orthogonal designs.Factorial designs Block-circulant matrices Auto-correlation