Related papers: New constructions of Hadamard matrices

In this paper we construct a new, previously unknown four-parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various sextic polynomials. We conjecture that the…

Let $n$ be the order of a (quaternary) Hadamard matrix. It is shown that the existence of a projective plane of order $n$ is equivalent to the existence of a balancedly multi-splittable (quaternary) Hadamard matrix of order $n^2$.

A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order $60$ with an automorphism of order $29$…

We construct new, previously unknown parametric families of complex conference matrices and of complex Hadamard matrices of square orders and related them to complex equiangular tight frames.

Hadamard matrices, orthogonal designs and amicable orthogonal designs have a number of applications in coding theory, cryptography, wireless network communication and so on. Product designs were introduced by Robinson in order to construct…

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…

We single out a class of difference families which is widely used in some constructions of Hadamard matrices and which we call Goethals--Seidel (GS) difference families. They consist of four subsets (base blocks) of a finite abelian group…

Complex Hadamard matrices have received considerable attention in the past few years due to their appearance in quantum information theory. While a complete characterization is currently available only up to order 5 (in \cite{haagerup}),…

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…

Perfect difference families (PDFs for short) are important both in theoretical and in applications. Perfect difference matrices (PDMs for short) and the equivalent structure had been extensively studied and used to construct perfect…

Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to $2$-designs, and have applications in constructions for…

Skew Hadamard difference sets are an interesting topic of study for over seventy years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in $\mathbb{F}_q$ where $q \equiv 3…

We continue our systematic search for symmetric Hadamard matrices based on the so called propus construction. In a previous paper this search covered the orders $4v$ with odd $v\le41$. In this paper we cover the cases $v=43,45,47,49,51$.…

A $Cretan(4t+1)$ matrix, of order $4t+1$, is an orthogonal matrix whose elements have moduli $\leq 1$. The only $Cretan(4t+1)$ matrices previously published are for orders 5, 9, 13, 17 and 37. This paper gives infinitely many new…

A complex Hadamard matrix is a square matrix W with complex entries of absolute value 1 satisfying WW*=nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we give constructions of complex…

Hadamard matrices of order $n$ are conjectured to exist whenever $n$ is $1$, $2$, or a multiple of $4$; a similar conjecture exists for skew Hadamard matrices. We provide constructions covering orders $\le 1208$ of all known Hadamard and…

\textit{Propus} (which means twins) is a construction method for orthogonal $\pm 1$ matrices based on a variation of the Williamson array called the \textit{propus array} \[ \begin{matrix*}[r] A& B & B & D B& D & -A &-B B& -A & -D & B D& -B…

Recently, Feng and Xiang \cite{FX113} found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple…

We study orthogonal matrices whose elements have moduli $\leq 1$. This paper shows that the existence of two such families of matrices is equivalent. Specifically we show that the existence of an Hadamard matrix of order $4t$ is equivalent…

An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $H\in M_N(\mathbb C)$ whose entries are on the unit…