Hadamard Matrices

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Jennifer Seberry - One of the best experts on this subject based on the ideXlab platform.

  • A new family of amicable Hadamard Matrices
    Journal of Statistical Theory and Practice, 2013
    Co-Authors: Jennifer Seberry
    Abstract:

    We study constructions for amicable Hadamard Matrices. The family for orders , t a positive integer, is explicitly exhibited. We also show that there are amicable Hadamard Matrices of order for any odd integer . Now we have orders , , , , …, an odd integer, for the first time.

  • On some applications of Hadamard Matrices
    Metrika, 2005
    Co-Authors: Jennifer Seberry, Beata Jwysocki, Tadeusz Awysocki
    Abstract:

    Modern communications systems are heavily reliant on statistical techniques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is Hadamard Matrices. They are used in many different ways and some examples are given. This paper concentrates on code division multiple access systems where Hadamard Matrices are used for user separation. Two older techniques from design and analysis of experiments which rely on similar processes are also included. We give a short bibliography (from the thousands produced by a google search) of applications of Hadamard Matrices appearing since the paper of Hedayat and Wallis in 1978 and some applications in telecommunications.

  • Hadamard Matrices, orthogonal designs and construction algorithms
    Designs 2002, 2003
    Co-Authors: Stelios Georgiou, Christos Koukouvinos, Jennifer Seberry
    Abstract:

    We discuss algorithms for the construction of Hadamard Matrices. We include discussion of construction using Williamson Matrices, Legendre pairs and the discret Fourier transform and the two circulants construction.Next we move to algorithms to determine the equivalence of Hadamard Matrices using the profile and projections of Hadamard Matrices. A summary is then given which considers inequivalence of Hadamard Matrices of orders up to 44.The final two sections give algorithms for constructing orthogonal designs, short amicable and amicable sets for use in the Kharaghani array.

  • The excess of complex Hadamard Matrices
    Graphs and Combinatorics, 1993
    Co-Authors: Hadi Kharaghani, Jennifer Seberry
    Abstract:

    A complex Hadamard matrix,C, of ordern has elements 1, −1,i, −i and satisfiesCC*=nInwhereC* denotes the conjugate transpose ofC. LetC=[cij] be a complex Hadamard matrix of order\(n. S(C) = \sum\limits_{ij} {c_{ij} } \) is called the sum ofC. σ(C)=|S(C)| is called the excess ofC. We study the excess of complex Hadamard Matrices. As an application many real Hadamard Matrices of large and maximal excess are obtained.

  • Product of four Hadamard Matrices
    Journal of Combinatorial Theory Series A, 1992
    Co-Authors: R. Craigen, Jennifer Seberry, Xian-mo Zhang
    Abstract:

    We prove that if there exist Hadamard Matrices of order 4m, 4n, 4p, and 4q then there exists an Hadamard matrix of order 16mnpq. This improves and extends the known result of Agayan that there exists a Hadamard matrix of order 8mn if there exist Hadamard Matrices of order 4m and 4n.

Udaya Parampalli - One of the best experts on this subject based on the ideXlab platform.

  • A class of quaternary noncyclic Hadamard Matrices
    The Australasian Journal of Combinatorics, 2014
    Co-Authors: Udaya Parampalli, Serdar Boztas
    Abstract:

    A normalized Hadamard matrix is said to be completely noncyclic if no two row vectors are shift equivalent in its punctured matrix (i.e., with the first column removed). In this paper we present an infinite recursive construction for completely noncyclic quaternary Hadamard Matrices. These Hadamard Matrices are useful in constructing low correlation zone sequences.

  • On the Noncyclic Property of Sylvester Hadamard Matrices
    IEEE Transactions on Information Theory, 2010
    Co-Authors: Xiaohu Tang, Udaya Parampalli
    Abstract:

    In this paper, we are concerned with Hadamard Matrices with a certain noncyclic property. First we show that when the first column of a Sylvester Hadamard matrix of order 2m, m ≥ 2, a positive integer, is removed, the number of shift distinct row vectors in the matrix is given by 2m-m. Then, for m ≥ 4, we construct an infinite family of Hadamard Matrices with a property that when the first column of the Hadamard matrix is removed, all the row vectors of the matrix are shift distinct. These Hadamard Matrices are useful in constructing low correlation zone sequences.

Hadi Kharaghani - One of the best experts on this subject based on the ideXlab platform.

  • Balancedly splittable Hadamard Matrices
    arXiv: Combinatorics, 2018
    Co-Authors: Hadi Kharaghani, Sho Suda
    Abstract:

    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.

  • mutually unbiased bush type Hadamard Matrices and association schemes
    arXiv: Combinatorics, 2014
    Co-Authors: Hadi Kharaghani, Sara Sasani, Sho Suda
    Abstract:

    It was shown by LeCompte, Martin, and Oweans in 2010 that the existence of mutually unbiased Hadamard Matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a $Q$-polynomial association scheme of class four which is both $Q$-antipodal and $Q$-bipartite. We prove that the existence of a set of mutually unbiased Bush-type Hadamard Matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain the upper bound of the number of mutually unbiased Bush-type Hadamard Matrices of order $4n^2$ to be $2n-1$. This is in contrast to the fact that the upper bound of mutually unbiased Hadamard Matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class $4$.

  • Hadamard Matrices of Order 32
    Journal of Combinatorial Designs, 2012
    Co-Authors: Hadi Kharaghani, Behruz Tayfeh-rezaie
    Abstract:

    Two Hadamard Matrices are considered equivalent if one is obtained from the other by a sequence of operations involving row or column permutations or negations. We complete the classification of Hadamard Matrices of order 32. It turns out that there are exactly 13,710,027 such Matrices up to equivalence.

  • Unbiased complex Hadamard Matrices and bases
    Cryptography and Communications, 2010
    Co-Authors: Darcy Best, Hadi Kharaghani
    Abstract:

    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.

  • On the classification of Hadamard Matrices of order 32
    Journal of Combinatorial Designs, 2010
    Co-Authors: Hadi Kharaghani, Behruz Tayfeh-rezaie
    Abstract:

    All equivalence classes of Hadamard Matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard Matrices. We flnd all equivalence classes of Hadamard Matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard Matrices of one type and 26,369 such Matrices of another type. Based on experience with the classiflcation of Hadamard Matrices of smaller order, it is expected that the number of the remaining two types of these Matrices, relative to the total number of Hadamard Matrices of order 32, to be insigniflcant.

Behruz Tayfeh-rezaie - One of the best experts on this subject based on the ideXlab platform.

  • Hadamard Matrices with few distinct types
    Linear and Multilinear Algebra, 2018
    Co-Authors: A. Mohammadian, Behruz Tayfeh-rezaie
    Abstract:

    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 resu...

  • Hadamard Matrices with few distinct types
    arXiv: Combinatorics, 2016
    Co-Authors: A. Mohammadian, Behruz Tayfeh-rezaie
    Abstract:

    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.

  • Hadamard Matrices of Order 32
    Journal of Combinatorial Designs, 2012
    Co-Authors: Hadi Kharaghani, Behruz Tayfeh-rezaie
    Abstract:

    Two Hadamard Matrices are considered equivalent if one is obtained from the other by a sequence of operations involving row or column permutations or negations. We complete the classification of Hadamard Matrices of order 32. It turns out that there are exactly 13,710,027 such Matrices up to equivalence.

  • On the classification of Hadamard Matrices of order 32
    Journal of Combinatorial Designs, 2010
    Co-Authors: Hadi Kharaghani, Behruz Tayfeh-rezaie
    Abstract:

    All equivalence classes of Hadamard Matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard Matrices. We flnd all equivalence classes of Hadamard Matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard Matrices of one type and 26,369 such Matrices of another type. Based on experience with the classiflcation of Hadamard Matrices of smaller order, it is expected that the number of the remaining two types of these Matrices, relative to the total number of Hadamard Matrices of order 32, to be insigniflcant.

Pingzhi Fan - One of the best experts on this subject based on the ideXlab platform.

  • Constructions of Factorizable Multilevel Hadamard Matrices
    IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 2009
    Co-Authors: S. Matsufuji, Pingzhi Fan
    Abstract:

    Factorization of Hadamard Matrices can provide fast algorithm and facilitate efficient hardware realization. In this letter, constructions of factorizable multilevel Hadamard Matrices, which can be considered as special case of unitary Matrices, are inverstigated. In particular, a class of ternary Hadamard Matrices, together with its application, is presented.

  • Construction of multilevel Hadamard Matrices with small alphabet
    Electronics Letters, 2008
    Co-Authors: Q.k. Trinh, Pingzhi Fan
    Abstract:

    Multilevel Hadamard Matrices, which are related to orthogonal design and binary Hadamard Matrices, are investigated. It is assumed that all elements of the matrix are integers. Construction of multilevel Hadamard Matrices with a small alphabet, such as three, four or eight different elements, is presented.

  • On the Existence of Multilevel Hadamard Matrices with Odd Order
    2007 3rd International Workshop on Signal Design and Its Applications in Communications, 2007
    Co-Authors: Q.k. Trinh, Pingzhi Fan, E.m. Gabidulin, R.n. Mohan
    Abstract:

    The multilevel (n-ary) Hadamard Matrices, which are related to orthogonal Matrices or orthogonal designs, are investigated in this paper. It is assumed that all matrix elements are integers, which make the binary Hadamard Matrices as special cases of multilevel (n-ary) Hadamard Matrices. It is shown that the multilevel Hadamard Matrices of odd order do exist if only two different matrix elements are contained; but, except an unknown case, multilevel (n-ary) Hadamard Matrices of odd order do not exist when all matrix elements are successive integers.

  • multilevel Hadamard Matrices and zero correlation zone sequences
    Electronics Letters, 2006
    Co-Authors: Q.k. Trinh, Pingzhi Fan, E.m. Gabidulin
    Abstract:

    Constructions of multilevel Hadamard Matrices are presented. Based on the multilevel Hadamard Matrices, multilevel zero correlation zone sequences which are useful in quasi-synchronous CDMA systems and other applications are derived.