Related papers: Three-dimensional Hadamard matrices of Paley type
In this paper we provide a general method to construct four-parameter families of complex Hadamard matrices of order six. Our approach is to write a 6-dimensional matrix as composed of four blocks, each one in the form of a circulant…
All generalized Hadamard matrices of order 18 over a group of order 3, H(6,3), are enumerated in two different ways: once, as class regular symmetric (6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of…
We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of…
This paper has an expository nature. We compare the spectral properties (such as boundedness and compactness) of three families of semi-infinite matrices and point out similarities between them. The common feature of these families is that…
We study the dimensions of Hadamard products of $k\ge 3$ varieties if we allow to modify $k-1$ of them by the action of a general projective linear transformation. We also prove that the join of a variety not contained in a coordinate…
We study the existence and construction of circulant matrices $C$ of order $n\geq2$ with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ and mutually orthogonal rows. These matrices generalize circulant conference ($d=0$) and…
Six non-affine 3-parameter families of complex Hadamard matrices of order 8 are presented. These families contain Hadamard matrices that are not equivalent to any previously known Hadamard matrices in the literature. Each family arises from…
The Hadamard maximal determinant problem asks for the largest n-by-n determinant with entries in {+1,-1}. When n is congruent to 1 (mod 4), the maximal excess construction of Farmakis & Kounias has been the most successful general method…
We construct explicitly groups associated to specific ternary algebras which extend the Lie (super)algebras (called Lie algebras of order three). It turns out that the natural variables which appear in this construction are variables which…
Let $q$ be a prime power such that $q\equiv 1\pmod{4}$. The Paley graph of order $q$ is the graph with vertex set as the finite field $\mathbb{F}_q$ and edges defined as, $ab$ is an edge if and only if $a-b$ is a non-zero square in…
An analytical method for getting new complex Hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided. The method is illustrated with the n=4 case that leads to a rich family of eight-dimensional…
In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices,…
A simple and algorithmic description of matrix shape invariant potentials is presented. The complete lists of generic matrix superpotentials of dimension $2\times2$ and of special superpotentials of dimension $3\times3$ are given…
The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. In this paper, we investigate Hadamard matrices with few distinct types. Among other results, the Sylvester Hadamard matrices are…
We provide a classification of the homogeneous 3-dimensional permutation structures, i.e. homogeneous structures in a language of 3 linear orders, partially answering a question of Cameron. We also arrive at a natural description of all…
In this paper we modify a fundamental block construction of Kharaghani and Seberry and show how to use certain circulant $\{-1,1\}$-matrices of odd order $p$ to construct a complex Hadamard matrix of order $2p$. In particular, for $p=47$ we…
In this paper we provide an analytical procedure which leads to a system of $(n-2)^2$ polynomial equations whose solutions give the parameterisation of the complex $n\times n$ Hadamard matrices. It is shown that in general the Hadamard…
A novel family of integrable third order maps is presented. Each map possesses, by construction, a pair of rational invariants and a commuting map from the same class. The 3-dimensional invariant curve is parametrized, in general, by an…
Matrices are very popular and widely used in mathematics and other fields of science. Every mathematician has known the properties of finite-sized matrices since the time of study. In this paper, we consider the basic theory of infnite…
A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on non zero complex parameters. The method we use is via…