The purpose of this paper is to introduce new parametric families of complex Hadamard matrices in two different ways. First, we prove that every real Hadamard matrix of order N>=4 admits an affine orbit. This settles a recent open problem of Tadej and Zyczkowski, who asked whether a real Hadamard matrix can be isolated among complex ones. In particular, we apply our construction to the only (up to equivalence) real Hadamard matrix of order 12 and show that the arising affine family is different from all previously known examples. Second, we recall a well-known construction related to real conference matrices, and show how to introduce an affine parameter in the arising complex Hadamard matrices. This leads to new parametric families of orders 10 and 14. An interesting feature of both of our constructions is that the arising families cannot be obtained via Dita's general method. Our results extend the recent catalogue of complex Hadamard matrices, and may lead to direct applications in quantum-information theory.
This paper is concerned with quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices, which are generalizations of unbiased Hadamard matrices, equivalently unbiased bases. These matrices are studied from the viewpoint of coding theory. As a consequence of a coding-theoretic approach, we provide upper bounds on the number of mutually quasi-unbiased Hadamard matrices. We give classifications of a certain class of self-complementary codes for modest lengths. These codes give quasi-unbiased Hadamard matrices and weakly unbiased Hadamard matrices. Some modification of the notion of w…
We will show that in a space of dimension $m$, any family of $2^{m-1}$ distinct Hadamard vectors (where you can choose x or -x but not both) can be partitioned into Hadamard matrices if and only if $m=2^n$ for some n. We will solve this problem with a simple algorithm for assigning the vectors to the Hadamard matrices.
Using the ideas of concatenation construction of codes over the $q$-ary alphabet, we modify the known generalized Sylvester-type construction of the Hadamard matrices. The new construction is based on two collections of the Hadamard matrices. In particular this construction involves $m$ Hadamard matrices of order $k$ and $k$ Hadamard matrices of order $m$. These matrices are not necessary different. As a result we obtain a Hadamard matrix of order $km$. The new construction gives many possibilities for construction of the new Hadamard matrices with different ranks and dimension of kernel.
In this article, a series of Hadamard matrix has been developed using some block matrices with the help of skew Hadamard matrix. Basically an internal structure of skew Hadamard matrix has been changed with some block matrices using kronecker product. For some parameter, Hadamard matrices of order 4t where t is an integer, has been found.
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.
We construct Hadamard matrices of orders 4x251 = 1004 and 4x631 = 2524, and skew-Hadamard matrices of orders 4x213 = 852 and 4x631 = 2524. As far as we know, such matrices have not been constructed previously. The constructions use the Goethals-Seidel array, suitable supplementary difference sets on a cyclic group and a new efficient matching algorithm based on hashing techniques.
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 shown to be characterized by their spectrum of types.
We define Bush-type generalized Hadamard matrices over abelian groups and construct symmetric Bush-type generalized Hadamard matrices over the additive group of finite field $\mathbb{F}_q$, $q$ a prime power. We then show and study an association scheme obtained from such generalized Hadamard matrices.
Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed.
We introduce mutually unbiased complex Hadamard (MUCH) matrices and show that the number of MUCH matrices of order 2n, n odd, is at most 2 and the bound is attained for n = 1,5,9. Furthermore, we prove that certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. As a consequence we generate a new pair of unbiased real Hadamard matrices of order 36.
Quaternary unit Hadamard (QUH) matrices were introduced by Fender, Kharagani and Suda along with a method to construct them at prime power orders. We present a novel construction of real Hadamard matrices from QUH matrices. Our construction recovers the result by Mukhopadhyay on the existence of real Hadamard matrices of order $q^n+q^{n-1}$ for each prime power $q\equiv 3\mod 4$. Furthermore we provide nonexistence conditions for QUH matrices.
We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the $7$-modular and $11$-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjecture for a sufficient condition for the existence of a $p$-modular Hadamard matrix for all but finitely many cases. When $2$ is a primitive root of a prime $p$, we conditionally solve this conjecture and therefore the $p$-modular version of the Hadamard conjecture for all but finitely many cases when $p \equiv 3 \pmod{4}$, and prove a weaker result for…
In this paper we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n != 6, 11. In particular, this solves the 5-modular version of the Hadamard conjecture.
Over forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph $X$ contains a Hadamard matrix then $X$ is either of Latin square type or of negative Latin square type. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency algebras. Moreover we show that there are only three distance regular covers of the complete graph that give complex Hadamard matrices.
We prove a conjecture due to Wanless about the permanent of Hadamard matrices in the particular case of Sylvester-Hadamard matrices. Namely we show that for all n greater or equal to 2, the dyadic valuation of the permanent of the Sylvester-Hadamard matrix of order n is equal to the dyadic valuation of n!. As a consequence, the permanent of the Sylvester-Hadamard matrix of order n doesn't vanish for n greater or equal to 2.
We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, F. Szöllősi, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.
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 to the existence of a Cretan$(4t-1,2)$-Mersenne matrix.
We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard matrices (MUHs) in any dimension $d$ cannot contain more than one real Hadamard matrix. We also give new proofs of several known structural results in low dimensions, for $d\le 6$.
We systematically explore the new method of construction (known as the propus construction) of symmetric Hadamard matrices for small orders, $4v$. In particular we give the first examples of symmetric Hadamard matrices of order $156=4\cdot 39$. The smallest $v$ for which symmetric Hadamard matrices of order $4v$ are not known is now $v=47$.
We discuss an extension of the almost Hadamard matrix formalism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices.
In this note, we study the existence of Hadamard matrices of order $36$ formed by codewords of weight $36$ in some ternary near-extremal self-dual codes of length $36$.
We introduce a class of regular unit Hadamard matrices whose entries consist of two complex numbers and their conjugates for a total of four complex numbers. We then show that these matrices are contained in the Bose-Mesner algebra of an association scheme arising from skew Paley matrices.
In this paper we classify complex Hadamard matrices contained in the Bose-Mesner algebra of nonsymmetric 3-class association schemes. As a consequence of our classification, we have two infinite families and some small examples of complex Hadamard matrices contained in the Bose-Mesner algebra of a self-dual fission of a complete multipartite graph.
In this paper, we find regular or biregular Hadamard matrices with maximum excess by negating some rows and columns of known Hadamard matrices obtained from quadratic residues of finite fields. In particular, we show that if either $4m^2+4m+3$ or $2m^2+2m+1$ is a prime power, then there exists a biregular Hadamard matrix of order $n=4(m^2+m+1)$ with maximum excess. Furthermore, we give a sufficient condition for Hadamard matrices obtained from quadratic residues being transformed to be regular in terms of four-class translation association schemes on finite fields.
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 $n^2$.
We give some very interesting matrices which are orthogonal over groups and, as far as we know, referenced, but in fact undocumented. This note is not intended to be published but available for archival reasons.
A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10x10 and 12x12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14x14 matrix that there cannot be a general method for introducing parameters into these types of matrices.
A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order $n$ all trades contain at least $n$ entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar $c \neq 1$ to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order $n$ and show that they all contain at least $n$ entries. We conjecture that all trades in complex Hadamard matrices contain at least $n$ entries.
There are several well-known methods that one can use to construct Hadamard matrices from base sequences BS(m,n). In view of the recent classification of base sequences BS(n+1,n) for n <= 30, it may be of interest to show on an example how prolific these methods are. For that purpose we have selected the Hadamard matrices of order 60. By using these methods and the transposition map we have constructed 1759 nonequivalent Hadamard matrices of order 60.
We prove that a circulant Hadamard code of length $4n$ can always be seen as an HFP-code (Hadamard full propelinear code) of type $C_{4n}\times C_2$, where $C_2=\langle u\rangle$ or the same, as a cocyclic Hadamard code. We compute the rank and dimension of the kernel of these kind of codes.
In this paper we present new Hadamard matrices and related combinatorial structures. In particular, it is constructed 5202 inequivalent Hadamard matrices of order 36 as well as 180538 Hadamard symmetric designs with 35 points in addition to those structures that admit an automorphism of order 3. Consequently, there are at least 272116 Hadamard 3-designs with 36 points and 70 lines. We found that all Hadamard matrices constructed here are equivalent to a regular Hadamard matrix. This fact contributes to the conjecture that Hadamard matrices of order 36, and possibly those of order $4m^2$, are r…
In this paper a method for embedding cocyclic submatrices with ``large'' determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these determinants attain the largest possible value, we are embedding D-optimal designs. Applications to the pivot values that appear when Gaussian Elimination with complete pivoting is performed on these cocyclic Hadamard matrices are studied.
Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enable us to describe a polynomial ideal that characterizes the set of cocyclic Hadamard matrices over a fixed finite group $G$ of order $4t$. Nevertheless, the complexity of the computation of the reduced Gröbner basis of this ideal is $2^{O(t^2)}$, which is excessive even for very small orders. In order to improve the efficiency of this polynomial method, we tak…
We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity. We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian…
\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} \[ \begin{matrix*}[r] A& B & B & D B& D & -A &-B B& -A & -D & B D& -B & B &-A. \end{matrix*} \] This construction designed to find symmetric Hadamard matrices was originally based on circulant symmetric $\pm 1$ matrices, called \textit{propus matrices}. We also give another construction based on symmetric Williamson-type matrices. We give constructions to find symmetric propus-Hadamard matrices for 57 orders $4n$, $n < 200$ o…
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 $v=43,45,47,49,51$. The odd integers $v<120$ for which no symmetric Hadamard matrices of order $4v$ are known are the following: $$47,59,65,67,73,81,89,93,101,103,107,109,113,119.$$ By using the propus construction, we found several symmetric Hadamard matrices of order $4v$ for $v=47,73,113$.
We give an example of a BH(5; 5) matrix M satisfying that the eigenvalues of the unitary matrix $\frac{1}{\sqrt5}M$ are all primitive 10-th roots of unity and such that $\frac{1}{5}M^3$ is not a BH(5; 5) matrix. This example gives a negative answer to a conjecture proposed by R. Egan and P. O. Cathain.
Following the ideas of [AGG11] about Zt x Z2,2-cocyclic Hadamard matrices, we introduce the notion of diagram, which visually represents any set of coboundaries. Diagrams are a very useful tool for the description and the study of paths and intersections, as described in [AGG11]. Then, we will study four different operations on Zt x Z2,2-cocyclic matrices. These operations will be defined on the set of coboundaries defining the matrix, preserve the Hadamard character of the cocyclic matrices, and allow us to obtain new Hadamard matrices from old ones. We split the set of Hadamard matrices into…
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 in the Bose-Mesner algebra of a certain 4-class symmetric association scheme. Moreover, we determine the Nomura algebras to show that the resulting matrices are not decomposable into nontrivial generalized tensor products.
We study the isolated partial Hadamard matrices, under the assumption that the entries are roots of unity, or more generally, under the assumption that the combinatorics comes from vanishing sums of roots of unity. We first review the various conjectures on the subject, and then we present several new results, regarding notably the master Hadamard matrices, and the McNulty-Weigert construction. We discuss then the notion of isolation in some related contexts, of the magic unitary matrices, and of the quantum permutation groups, with a number of conjectures on the subject.
Correction of paper published in J. Combinatorial Theory 21, 1976: On the Existence of Hadamard Matrices
在J. Combinatorial Theory 21，1976中的On the Existence of Hadamard Matrices一文中，我们证明了对于一个自然数q > 3，我们可以构造一个2阶Hadamard矩阵q，其中s \geq t = [2 log_2（q-3）].我将表明，这一界是不是在文件中给出的证明的后果，并解释在论证中的错误。
