Complex Hadamard matrices and Equiangular Tight Frames

In this paper we give a new construction of parametric families of complex Hadamard matrices of square orders, and connect them to equiangular tight frames. The results presented here generalize some of the recent ideas of Bodmann et al. and extend the list of known equiangular tight frames. In particular, a (36,21) frame coming from a nontrivial cube root signature matrix is obtained for the first time.Comment: 6 pages, contribution to the 16th ILAS conference, Pisa, 201

A characterization of skew Hadamard matrices and doubly regular tournaments

We give a new characterization of skew Hadamard matrices of size $n$ in terms of the data of the spectra of tournaments of size $n-2$.Comment: 9 page

Supplementary difference sets with symmetry for Hadamard matrices

First we give an overview of the known supplementary difference sets (SDS) (A_i), i=1..4, with parameters (n;k_i;d), where k_i=|A_i| and each A_i is either symmetric or skew and k_1 + ... + k_4 = n + d. Five new Williamson matrices over the elementary abelian groups of order 25, 27 and 49 are constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127 are presented. The last of these is obtained from a (127,57,76)-difference family that we have constructed. An old non-published example of G-matrices of order 37 is also included.Comment: 16 pages, 2 tables. A few minor changes are made. The paper will appear in Operators and Matrice

Implementing Hadamard Matrices in SageMath

Hadamard matrices are $(-1, +1)$ square matrices with mutually orthogonal rows. The Hadamard conjecture states that Hadamard matrices of order $n$ exist whenever $n$ is $1$, $2$, or a multiple of $4$. However, no construction is known that works for all values of $n$, and for some orders no Hadamard matrix has yet been found. Given the many practical applications of these matrices, it would be useful to have a way to easily check if a construction for a Hadamard matrix of order $n$ exists, and in case to create it. This project aimed to address this, by implementing constructions of Hadamard and skew Hadamard matrices to cover all known orders less than or equal to $1000$ in SageMath, an open-source mathematical software. Furthermore, we implemented some additional mathematical objects, such as complementary difference sets and T-sequences, which were not present in SageMath but are needed to construct Hadamard matrices. This also allows to verify the correctness of the results given in the literature; within the $n\leq 1000$ range, just one order, $292$, of a skew Hadamard matrix claimed to have a known construction, required a fix.Comment: pdflatex+biber, 32 page

Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces

We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a $2$-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order $n$ exists, if there exists a skew Hadamard matrix of this size. This is the case for any even dimension $n\le 20$, and for these dimensions we demonstrate that a bistochastic matrix $B$ located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix $U$, such that $B_{ij}=|U_{ij}|^2$, is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order $n \times n$. Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis.Comment: 17 page