Related papers: On a class of quaternary complex Hadamard matrices
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…
Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup…
All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four…
We present an infinite number of construction schemes involving unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on biunitary…
We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic…
We study the partial Hadamard matrices $H\in M_{M\times N}(\mathbb C)$ which are regular, in the sense that the scalar products between pairs of distinct rows decompose as sums of cycles (rotated sums of roots of unity). The simplest…
The complete classification of $6\times 6$ complex Hadamard matrices (CHMs) is a long-standing open problem. In this paper we investigate a series of CHMs, such as the CHMs containing a $2\times 3$ submatrix with rank one, the CHMs…
A Hadamard matrix is balanced splittable if some subset of its rows has the property that the dot product of every two distinct columns takes at most two values. This definition was introduced by Kharaghani and Suda in 2019, although…
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…
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…
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. \'O Cath\'ain and R\"oder described a classification algorithm for CHMs of order $4n$ based on…
It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard matrix…
A Hadamard matrix $H$ of order $n$ is a square matrix with entries $\pm 1$ satisfying $HH^T = nI_n$, where $I_n$ is the identity matrix of order $n$. A circulant Hadamard matrix is a Hadamard matrix whose rows are cyclic shifts of one…
We show that the Nomura algebra of the nonsymmetric Hadamard model coincides with the Bose-Mesner algebra of the directed Hadamard graph.
We define generalised notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum latin squares and unitary error bases are all given by biunitary elements…
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…
In this paper we disprove the Haagerup statement that all complex Hadamard matrices of order five are equivalent with the Fourier matrix $F_5$ by constructing circulant matrices that lead to new Hadamard matrices. An important item is the…
New sets of rank n-representations of Temperley-Lieb algebra TL_N(q) are constructed. They are characterized by two matrices obeying a generalization of the complex Hadamard property. Partial classifications for the two matrices are given,…
We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This…
We have extended the Paley constructions for Hadamard matrices and obtained some series of Hadamard matrices. Especially Paley construction-II is applicable for odd prime power q is congruent to 1(mod 4) however our method is applicable for…