Symmetric Hadamard Research Articles

Two-level Cretan matrices constructed using SBIBD

Abstract Two-level Cretan matrices are orthogonal matrices with two elements, x and y. At least one element per row and column is 1 and the other element has modulus ≤ 1. These have been studied in the Russian literature for applications in image processing and compression. Cretan matrices have been found by both mathematical and computational methods but this paper concentrates on mathematical solutions for the first time. We give, for the first time, families of Cretan matrices constructed using the incidence matrix of a symmetric balanced incomplete block design and Hadamard related difference sets.

Conjugate Symmetric Discrete Orthogonal Transform

A new conjugate symmetric discrete orthogonal transform (CS-DOT)-generating method is proposed. The spectra of the CS-DOT for real input signals are conjugate symmetric so that we only need half memory size to store data. Meanwhile, the proposed CS-DOT also has a radix-2 fast algorithm so that it is suitable for hardware implementation. The CS-DOT generalized the existing transforms such that the discrete Fourier transform (DFT) and the conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) are special cases of the CS-DOT. The CS-DOT-generating method is more systematic and generalized than that of the original CS-SCHT. We can use the same implementation structure but only adjust the twiddle factors to construct the CS-SCHT and DFT so that it is easy to switch the behaviors between these transforms.

On the existence of tight relative 2-designs on binary Hamming association schemes

It is known that there is a close analogy between “Euclidean t-designs vs. spherical t-designs” and “Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs”. In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t-designs on p shells on a wide class of Q-polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t-designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2e-designs for e≥2. So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H(n,2). We prove that every tight relative 2-design on 2 shells in H(n,2) has the structure of coherent configuration. We determine all the possible parameters of coherent configurations attached to such tight relative 2-designs for n≤30. Moreover for each of them we determine whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n≡6(mod8), there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.

Divisible difference families from Galois rings $$GR(4,n)$$ GR ( 4 , n ) and Hadamard matrices

We give a new construction of difference families generalizing Szekeres's difference families Szekeres (Enseignment Math 15:269---278, 1969). As an immediate consequence, we obtain some new examples of difference families with several blocks in multiplicative subgroups of finite fields. We also prove that there exists an infinite family of divisible difference families with two blocks in a unit subgroup of the Galois ring $$GR(4,n)$$ GR ( 4 , n ) . Furthermore, we obtain a new construction method of symmetric Hadamard matrices by using divisible difference families and a new array.

Sliding Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform: Fast Algorithm and Applications

This paper presents a fast algorithm for the computation of sliding conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT). The algorithm calculates the values of window <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> /4 from those of window <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> , one length- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> /4 Walsh Hadamard transform (WHT) and one length- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> /4 Modified WHT (MWHT). The proposed algorithm requires O( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) arithmetic operations, which is more efficient than the block-based algorithms of various transforms and the sliding FFT algorithm, but less efficient than the sliding WHT algorithms. Compared to the recently proposed sliding inverse SCHT (ISCHT) algorithm, the proposed algorithm is more efficient for real input but less efficient for complex input. The applications of the sliding CS-SCHT in transform domain adaptive filtering (TDAF) to complex signal channel equalization and real speech signal acoustic echo cancellation are also provided.

Complete minimal graphs with prescribed asymptotic boundary on rotationally symmetric Hadamard surfaces

It is proved that if M is a rotationally symmetric Hadamard surface which is conformally equivalent to the hyperbolic disk then the asymptotic Dirichlet problem for the minimal surface equation is uniquely solvable for any continuous asymptotic boundary data. This result gives a partial answer of a question in Galvez and Rosenberg (Am J Math 132:1249–1273, 2010) about the existence of entire minimal graphs on Hadamard surfaces with sectional curvature possibly degenerating at infinity.

Divisible design graphs

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of ( v , k , λ ) -graphs, and like ( v , k , λ ) -graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as ( v , k , λ ) -graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.

Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

A new transform known as conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) is presented in this paper. The transform matrix of this transform possesses sequency ordering and the spectrum obtained by the CS-SCHT is conjugate symmetric. Some of its important properties are discussed and analyzed. Sequency defined in the CS-SCHT is interpreted as compared to frequency in the discrete Fourier transform. The exponential form of the CS-SCHT is derived, and the proof of the dyadic shift invariant property of the CS-SCHT is also given. The fast and efficient algorithm to compute the CS-SCHT is developed using the sparse matrix factorization method and its computational load is examined as compared to that of the SCHT. The applications of the CS-SCHT in spectrum estimation and image compression are discussed. The simulation results reveal that the CS-SCHT is promising to be employed in such applications.

Towards a Classification of 6 × 6 Complex Hadamard Matrices

Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţă's matrix D6. This answers a recent question of Bengtsson et al. [3].

Symmetric Bush-type Hadamard matrices of order $4m^4$ exist for all odd $m$

Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order 4m 4 for all odd integers m.

The Existence of a Bush-Type Hadamard Matrix of Order 324 and Two New Infinite Classes of Symmetric Designs

A symmetric 2-(324, 153, 72) design is constructed that admits a tactical decomposition into 18 point and block classes of size 18 such that every point is in either 0 or 9 blocks from a given block class, and every block contains either 0 or 9 points from a given point class. This design is self-dual and yields a symmetric Hadamard matrix of order 324 of Bush type, being the first known example of a symmetric Bush-type Hadamard matrix of order 4n^2 for n > 1 odd. Equivalently, the design yields a strongly regular graph with parameters v=324, k=153, \lambda=\mu=72 that admits a spread of cocliques of size 18. The Bush-type Hadamard matrix of order 324 leads to two new infinite classes of symmetric designs with parameters v=324(289^m+289^{m-1}+\cdots+289+1), \quad k=153(289)^m, \quad \lambda=72(289)^m, and v=324(361^m+361^{m-1}+\cdots+361+1), \quad k=171(361)^m, \quad \lambda=90(361)^m, where m is an arbitrary positive integer.

Bounds on the Number of Affine, Symmetric, and Hadamard Designs and Matrices

Lower bouds on the number of non-isomorphic embeddings of a symmetric net into affine designs with classical parameters, of an affine design into symmetric designs with classical parameters, and of a symmetric Hadamard design of order n into ones of order 2n are obtained. The bound of Jungnickel on the number of affine 2-(qd, qd−1, (qd−1−1)/(q−1)) designs (d⩾3) that contain the classical (q, qd−2)-net is improved by a factor of q3+4+…+d(q−1)d−2. Similarly, the bound of Jungnickel for the number of symmetric 2-((qd+1−1)/(q−1), (qd−1)/(q−1), (qd−1−1)/(q−1)) designs (d⩾3) that contain the the classical affine design AG(d, q) as a residual design is improved to match that of Kantor. Furthermore, for d large and by starting with rigid symmetric and affine designs, the lower bound for the number of non-isomorphic symmetric 2-((qd+1−1)/(q−1), (qd−1)/(q−1), (qd−1−1)/(q−1)) designs is improved to (qd−1+…+q)!. By using the Paley design of order n=(q+1)/4, q≡3 (mod4) a prime power, a lower bound for the number of Hadamard designs of order q+1 is also obtained. In particular, by choosing a non-classical net and non-classical affine design as the starting point, the bound on the number of symmetric 2-(40, 13, 4) designs is improved from 389 to 1, 108, 800, and the bound on the number of affine 2-(64, 16, 5) designs is improved from 157 to 10, 810, 800. A similar method also improves the number of non-isomorphic Hadamard 2-(31, 15, 7) designs from 1, 266, 891 to 11, 727, 788 and the number of non-isomorphic Hadamard 2-(39, 19, 9) designs from 38 to 5.87×1014. The number of inequivalent Hadamard matrices of order 40 is at least 3.66×1011.

Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles

Using cohomology we show that in studying the existence of an abelian non-splitting (4 t, 2, 4 t, 2 t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets.

Symmetric Versus Non-Symmetric Spin Models for Link Invariants

We study spin models as introduced in [20]. Such a spin model can be defined as a square matrix satisfying certain equations, and can be used to compute an associated link invariant. The link invariant associated with a symmetric spin model depends only trivially on link orientation. This property also holds for quasi-symmetric spin models, which are obtained from symmetric spin models by certain “gauge transformations” preserving the associated link invariant. Using a recent result of [16] which asserts that every spin model belongs to some Bose-Mesner algebra with duality, we show that the transposition of a spin model can be realized by a permutation of rows. We call the order of this permutation the index of the spin model. We show that spin models of odd index are quasi-symmetric. Next, we give a general form for spin models of index 2 which implies that they are associated with a certain class of symmetric spin models. The symmetric Hadamard spin models of [21] belong to this class and this leads to the introduction of non-symmetric Hadamard spin models. These spin models give the first known example where the associated link invariant depends non-trivially on link orientation. We show that a non-symmetric Hadamard spin model belongs to a certain triply regular Bose-Mesner algebra of dimension 5 with duality, and we use this to give an explicit formula for the associated link invariant involving the Jones polynomial.

Hadamard matrices and spin models

The concept of spin models was introduced by Jones in 1989. Kawagoe, Munemasa and Watatani generalized it by removing the condition of symmetry. Recently, Bannai and Bannai further generalized the concept of spin models which is called 4-weight spin models or generalized generalized spin models. On the other hand, Ivanov and Chuvaeva showed that symmetric amorphous association schemes of class 4 obtained from Hadamard matrices. An infinite family of Hadamard matrices and of complex Hadamard matrices can be constructed by fusing the relations of these amorphous association schemes. We show the necessary and sufficient condition that these Hadamard matrices give generalized spin models of symmetric Hadamard type and of pseudo-Jones type. A special class of Hadamard matrices satisfies this necessary and sufficient condition. Furthermore, Hadamard matrices constructed from amorphous association schemes are also contained in the special class if Hadamard matrices giving these amorphous association schemes are contained in the special class. It means that there exist infinite families of generalized spin models of symmetric Hadamard type and of pseudo-Jones type.

Trace, Symmetry and Orthogonality

AbstractDoes there exist a circulant conference matrix of order &gt; 2? When is there a symmetric Hadamard matrix with constant diagonal? How many pairwise disjoint, amicable weighing matrices of order n can there be? These are questions concerning which the trace function gives a great deal of insight. We offer easy proofs of the known solutions to the first two, the first being new, and develop new results regarding the latter question. It is shown that there are 2t disjoint amicable weighing matrices of order 2tp, where p is odd, and that this is an upper bound for t ≤ 1. An even stronger bound is obtained for certain cases.

Renormalized graviton stress-energy tensor in curved vacuum space-times.

The graviton contribution to the renormalized one-loop effective stress-energy tensor is calculated in an arbitrary vacuum space-time with a cosmological constant. This is done for (1) the standard definition of the effective action and (2) the reparametrization-invariant effective action of Vilkovisky and DeWitt. The renormalized one-loop effective stress-energy tensor is given in terms of the graviton two-point function, by making use of its symmetric Hadamard representation.

On a theorem of Brualdi and Newman

Using the Fourier transform, we give an easy and conceptual proof of a theorem of Brualdi and Newman which states that there are no symmetric circulant Hadamard matrices of order greater than 4.

Photon propagators and the definition and approximation of renormalized stress tensors in curved space-time.

We present the symmetric Hadamard representation for scalar and photon Feynman Green's functions. We use these representations to give a simple definition for their associated renormalized stress tensors. We investigate the connection between the accuracy of the WKB approximation and the vanishing of the trace anomaly for these fields. We show that, although for scalars there is a direct connection, this is not true for photons, and we discuss the relevance of these results to the approximation of renormalized stress tensors in static Einstein space-times.

Symmetric Hadamard series

In a general curved space–time, the requirements that the Feynman Green’s function be symmetric and have the Hadamard form are shown to result in specific constraints on the local behavior of the function. These constraints are solved yielding a general form for the function.