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Showing papers on "Hadamard transform published in 2006"


Journal ArticleDOI
TL;DR: The algorithm can be used to compile Shor's algorithm into an efficient fault-tolerant form using only Hadamard, controlled-not, and π/8 gates, and is generalized to apply to multi-qubit gates and togates from SU(d).
Abstract: This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form ofan efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequenceof gates from a fixed and finite set. The algorithm can be used, for example, to compileShor's algorithm, which uses rotations of π/2k, into an efficient fault-tolerant form usingonly Hadamard, controlled-not, and π/8 gates. The algorithm runs in O(log2.71(1/e))time, and produces as output a sequence of O(log3.97(1/e)) quantum gates which isguaranteed to approximate the desired quantum gate to an accuracy within e > 0. Wealso explain how the algorithm can be generalized to apply to multi-qubit gates and togates from SU(d).

317 citations


Journal ArticleDOI
TL;DR: Basic properties of complex Hadamard matrices are reviewed and a catalogue of inequivalent cases known for the dimensions N = 2, 16, 12, 14 and 16 are presented.
Abstract: Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices and present a catalogue of inequivalent cases known for the dimensions N = 2,..., 16. In particular, we explicitly write down some families of complex Hadamard matrices for N = 12,14 and 16, which we could not find in the existing literature.

284 citations


Journal ArticleDOI
TL;DR: A family of new perfect nonlinear functions are presented, and it is shown that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamards difference sets.

259 citations


Journal Article
TL;DR: In this paper, it was shown that the Spectral Set Conjecture is false for all sets A of size |A| = 5, in any finite Abelian group.
Abstract: By analyzing the connection between complex Hadamard matrices and spectral sets, we prove the direction ?spectral ) tile? of the Spectral Set Conjecture, for all sets A of size |A|  5, in any finite Abelian group. This result is then extended to the infinite grid Zd for any dimension d, and finally to Rd. It was pointed out recently in [16] that the corresponding statement fails for |A| = 6 in the group Z5 3, and this observation quickly led to the failure of the Spectral Set Conjecture in R5 [16], and subsequently in R4 [13]. In the second part of this note we reduce this dimension further, showing that the direction ?spectral ) tile? of the Spectral Set Conjecture is false already in dimension 3.

137 citations


Journal ArticleDOI
TL;DR: In this paper, the Hadamard transform was applied to the ion gate of an atmospheric pressure ion mobility spectrometer fitted with an electrospray ionization source to produce a pseudorandom sequence of 1023 points.
Abstract: A detection scheme that makes use of the Hadamard transform has been employed with an atmospheric-pressure ion mobility spectrometer fitted with an electrospray ionization source. The Hadamard transform was implemented through the use of a linear-feedback shift register to produce a pseudorandom sequence of 1023 points. This pseudorandom sequence was applied to the ion gate of the spectrometer, and deconvolution of the ion signal was accomplished by the Hadamard transform to reconstruct the mobility spectrum. Ion mobility spectra were collected in both a conventional and Hadamard mode, with comparisons made between the two approaches. Initial results exhibited low spectral definition, so an oversampling technique was applied to increase the number of data points across each analyte spectral peak. The use of the Hadamard transform increases the duty cycle of the instrument to 50% and results in a roughly 5-fold enhancement of the signal-to-noise ratio with a negligible loss of instrument resolution. It is ...

120 citations


Journal ArticleDOI
TL;DR: In this paper, the authors improved the DeWitt-Schwinger and Hadamard representations of the Feynman propagator of a massive scalar field theory defined on an arbitrary gravitational background by deriving higher-order terms for the covariant Taylor series expansions of the geometrical coefficients.
Abstract: Having in mind applications to gravitational wave theory (in connection with the radiation reaction problem), stochastic semiclassical gravity (in connection with the regularization of the noise kernel) and quantum field theory in higher-dimensional curved spacetime (in connection with the Hadamard regularization of the stress-energy tensor), we improve the DeWitt-Schwinger and Hadamard representations of the Feynman propagator of a massive scalar field theory defined on an arbitrary gravitational background by deriving higher-order terms for the covariant Taylor series expansions of the geometrical coefficients -- i.e., the DeWitt and Hadamard coefficients -- that define them.

102 citations



Journal ArticleDOI
TL;DR: All the possible nonisomorphic additive Hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one.
Abstract: All the possible nonisomorphic additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one.

61 citations


Journal ArticleDOI
TL;DR: A Hadamard-Walsh code-based MC-CDMA system that achieves zero MAI over a frequency-selective fading channel and every user can enjoy a channel diversity gain of order L to improve the bit error performance.
Abstract: It is known that multicarrier code-division multiple-access (MC-CDMA) systems suffer from multiaccess interference (MAI) when the channel is frequency-selective fading. In this paper, we propose a Hadamard-Walsh code-based MC-CDMA system that achieves zero MAI over a frequency-selective fading channel. In particular, we will use appropriately chosen subsets of Hadamard-Walsh code as codewords. For a multipath channel of length L, we partition a Hadamard-Walsh code of size N into G subsets, where G is a power of two with GgesL. We will show that the N/G codewords in any of the G subsets yields an MAI-free system. That is, the number of MAI-free users for each codeword subset is N/G. Furthermore, the system has the additional advantage that it is robust to carrier frequency offset (CFO) in a multipath environment. It is also shown that the MAI-free property allows us to estimate the channel of each user separately and the system can perform channel estimation much more easily. Owing to the MAI-free property, every user can enjoy a channel diversity gain of order L to improve the bit error performance. Finally, we discuss a code priority scheme for a heavily loaded system. Simulation results are given to demonstrate the advantages of the proposed code and code priority schemes

61 citations


Journal ArticleDOI
TL;DR: This new technique offers a PAPR that is significantly lower than those of the best-known techniques without any loss in terms of energy and/or spectral efficiency, and without any side information being transmitted, and has a low computational complexity.
Abstract: In this paper, the problem of reducing the peak-to-average-power ratio (PAPR) in an orthogonal frequency-division multiplexing system is considered. We design a cubic constellation, called the Hadamard constellation, whose boundary is along the bases defined by the Hadamard matrix in the transform domain. Then, we further reduce the PAPR by applying the selective-mapping technique. The encoding method, following the method introduced in the work of Kwok, is derived from a decomposition known as the Smith normal form. This new technique offers a PAPR that is significantly lower than those of the best-known techniques without any loss in terms of energy and/or spectral efficiency, and without any side information being transmitted. Moreover, it has a low computational complexity.

57 citations


Journal ArticleDOI
TL;DR: In this article, the authors proposed a scheme to implement the discrete Hadamard walk in one dimension using a coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC).
Abstract: We propose a scheme to implement the simplest and best-studied version of the quantum random walk, the discrete Hadamard walk, in one dimension using a coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC). Implementation of the quantum walk using a BEC gives access to the familiar quantum phenomena on a macroscopic scale. This paper uses a rf pulse to implement the Hadamard operation (rotation) and stimulated Raman transition technique as a unitary shift operator. The scheme suggests the implementation of the Hadamard operation and unitary shift operator while the BEC is trapped in a long Rayleigh range optical dipole trap. The Hadamard rotation and a unitary shift operator on a BEC prepared in one of the internal states followed by a bit-flip operation, implements one step of the Hadamard walk. To realize a sizable number of steps, the process is iterated without resorting to intermediate measurement. With current dipole trap technology, it should be possible to implement enough steps to experimentally highlight the discrete quantum random walk using a BEC leading to further exploration of quantum random walks and its applications.

Journal ArticleDOI
TL;DR: A direct 2-D transform algorithm which suitably arranges the data processing sequences adopted in row and column transforms of H.264 CODEC systems to finish the data transposition on-the-fly to achieve the real-time multitransform processing of digital cinema video.
Abstract: This paper proposes a high-performance direct two-dimensional transform coding IP design for MPEG-4 AVC/H.264 video coding standard. Because four kinds of 4 /spl times/ 4 transforms, i.e., forward, inverse, forward-Hadamard, and inverse-Hadamard transforms are required in a H.264 encoding system, a high-performance multitransform accelerator is inevitable to compute these transforms simultaneously for fitting real-time processing requirement. Accordingly, this paper proposes a direct 2-D transform algorithm which suitably arranges the data processing sequences adopted in row and column transforms of H.264 CODEC systems to finish the data transposition on-the-fly. The induced new transform architecture greatly increases the data processing rate up to 8 pixels/cycle. In addition, an interlaced I/O schedule is disclosed to balance the data I/O rate and the data processing rate of the proposed multitransform design when integrated with H.264 systems. Using a 0.18-/spl mu/m CMOS technology, the optimum operating clock frequency of the proposed multitransform design is 100 MHz which achieves 800 Mpixels/s data throughput rate with the cost of 6482 gates. This performance can achieve the real-time multitransform processing of digital cinema video (4096 /spl times/ 4 2048@30 Hz). When the data throughput rate per unit area is adopted as the comparison index in hardware efficiency, the proposed design is at least 1.94 times more efficient than the existing designs. Moreover, the proposed multitransform design can achieve HDTV 720p, 1080i, digital cinema video processing requirements by consuming only 0.58, 2.91, and 24.18 mW when operated at 22, 50, and 100 MHz with 0.7, 1.0, and 1.8 V power supplies, respectively.

Journal ArticleDOI
TL;DR: A correlation-based scheme is used, but instead of gray values, it correlates numbers formulated by different combinations of the extracted local Walsh coefficients of the images, which forms a unified framework in terms of which several feature matching methods may be interpreted.
Abstract: This paper presents a new algorithm which can be used to register images of the same or different modalities, e.g., images with multiple channels, such as X-rays, temperature or elevation, or simply images of different spectral bands. In particular, a correlation-based scheme is used, but instead of gray values, it correlates numbers formulated by different combinations of the extracted local Walsh coefficients of the images. Each image patch is expanded in terms of Walsh basis functions. Each Walsh basis function can be thought of as measuring a different aspect of local structure, e.g., horizontal edge, corner, etc. The coefficients of the expansion, therefore, can be thought of as dense local features, estimating at each point the degree of presence of, for example, a horizontal edge, a corner with contrast of a certain type, etc. These coefficients are normalized and used as digits in a chosen number system which allows one to create a unique number for each type of local structure. The choice of the basis of the number system allows one to give different emphasis to different types of local feature (e.g., corners versus edges), and, thus, the method we present forms a unified framework in terms of which several feature matching methods may be interpreted. The algorithm is compared with wavelet and contour based approaches, using simulated and real images. The two images are assumed to differ from each other by a rotation and a translation only.

Journal Article
TL;DR: In this article, a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit and his signature theory of planar-match gate/matchgrid was established, which unified the two theories in expressibility.
Abstract: We establish a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [3], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plucker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.

Journal ArticleDOI
03 Feb 2006
TL;DR: Using reversible Hadamard difference sets, this article constructed a symmetric Bush-type hadamard matrices of order 4m 4 for all odd integers m, where m is the number of odd integers.
Abstract: Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order 4m 4 for all odd integers m.

Proceedings ArticleDOI
20 Aug 2006
TL;DR: A novel method to detect the moving cast shadows in the scene using normalized coefficients of orthogonal transform of image block are proved to be illumination invariant and are used to classify moving shadows and foreground objects.
Abstract: In many image and computer vision applications, shadows interfere with fundamental tasks such as moving objects segmentation and tracking. In this paper, a novel method is proposed to detect the moving cast shadows in the scene. The normalized coefficients of orthogonal transform of image block are proved to be illumination invariant and are used to classify moving shadows and foreground objects. Five kinds of orthogonal transform: DCT, DFT, Haar Transform, SVD and Hadamard Transform, are utilized in our work to detect moving cast shadows. Experimental results show that the proposed method succeeds in detecting moving cast shadows within indoor and outdoor environments.

Proceedings ArticleDOI
20 Aug 2006
TL;DR: The experimental results show the much improved performance of the proposed method in comparison with existing techniques, and also its robustness against the most common attacks.
Abstract: We present a new approach for transparent and high rate embedding of watermarks into digital images using fast Hadamard transform (FHT) and singular value decomposition (SVD). The proposed algorithm consists of three main steps: dividing the cover image into small blocks, applying the FHT to each block, and distributing the singular values of the visual watermark image over the transformed cover blocks. The main attractive features of this approach are simplicity, flexibility in data embedding capacity, and real-time implementation. The experimental results show the much improved performance of the proposed method in comparison with existing techniques, and also its robustness against the most common attacks.

Journal ArticleDOI
TL;DR: The concept of Hadamard ideal is introduced, to systematize the application of computational algebra methods to the construction of HadAmard matrices with two circulant cores, given by Fletcher, Gysin and Seberry.
Abstract: We apply computational algebra methods to the construction of Hadamard matrices with two circulant cores, given by Fletcher, Gysin and Seberry. We introduce the concept of Hadamard ideal, to systematize the application of computational algebra methods for this construction. We use the Hadamard ideal formalism to perform exhaustive search constructions of Hadamard matrices with two circulant cores for the twelve orders 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52. The total number of such Hadamard matrices is proportional to the square of the parameter. We use the Hadamard ideal formalism to compute the proportionality constants for the twelve orders listed above. Finally, we use the Hadamard ideal formalism to improve the lower bounds for the number of inequivalent Hadamard matrices for the seven orders 44, 48, 52, 56, 60, 64, 68.

Journal ArticleDOI
TL;DR: Integration of holograms into multimode waveguides allows the implementation of arbitrary unitary mode transformations and unitary matrix-vector multiplication.
Abstract: Integration of holograms into multimode waveguides allows the implementation of arbitrary unitary mode transformations and unitary matrix-vector multiplication. Theoretical analysis is used to justify a design approach to implement specific functions in these devices. Based on this approach, a compact mode-order converter, a Hadamard transformer, and a spatial pattern generator-correlator are proposed and analyzed. Beam propagation simulations are used to verify the theoretical calculations and to address bandwidth, scalability, and fabrication criteria. Optical pattern generators were successfully fabricated using standard photolithographic techniques to demonstrate the feasibility of the devices.

Journal ArticleDOI
TL;DR: The capability of fHTCE to perform time-resolved monitoring of changes in the concentration of model neurochemical compounds and improved signal-to-noise ratios with a total analysis time under 10 s is demonstrated.
Abstract: We report a new approach for collecting and deconvoluting the data in Hadamard transform capillary electrophoresis, referred to as fast Hadamard transform capillary electrophoresis (fHTCE). Using fHTCE, total analysis times can be reduced by up to 48% per multiplexed separation compared to conventional Hadamard transform capillary electrophoresis (cHTCE) while providing comparable signal-to-noise ratio enhancements. In fHTCE, the sample is injected following a pseudorandom pulsing sequence derived from the first row of a simplex matrix (S-matrix) in contrast to cHTCE, which utilizes a sequence of twice the length. In addition to the temporal savings provided by fHTCE, a 50% reduction in sample consumption is also realized due to the decreased number of sample injections. We have applied fHTCE to the analysis of mixtures of neurotransmitters and related compounds to yield improved signal-to-noise ratios with a total analysis time under 10 s. In addition, we demonstrate the capability of fHTCE to perform ti...

01 Jan 2006
TL;DR: A new code structure for spectral amplitude coding optical code division multiple access systems based on double weight (DW) code families is proposed and it has been observed that theoretical analysis and simulation for EDW is much better performance compared to Hadamard and Modified Frequency-Hopping codes.
Abstract: A new code structure for spectral amplitude coding optical code division multiple access systems based on double weight (DW) code families is proposed. The DW has a fixed weight of two. Enhanced double-weight (EDW) code is another variation of a DW code family that can has a variable weight greater than one. The EDW code possesses ideal cross-correlation properties and exists for every natural number n. A much better performance can be provided by using the EDW code compared to the existing code such as Hadamard and Modified Frequency-Hopping (MFH) codes. It has been observed that theoretical analysis and simulation for EDW is much better performance compared to Hadamard and Modified Frequency-Hopping (MFH) codes.

Journal ArticleDOI
TL;DR: In this article, the first eigenvalue of the Dirichlet-Laplacian in three domains (C1, 1-domains, Lipschitz domains, and bounded domains) was considered.

Journal Article
TL;DR: In this article, a complete list of Boolean symmetric signatures over bases of size 1 was given, and it was shown that the same unexpected holographic algorithm can be realized over a basis of size 2.
Abstract: In holographic algorithms, symmetric signatures have been particularly useful. We give a complete characterization of these symmetric signatures over all bases of size 1. These improve previous results by Cai and Choudhary (ICALP 2006, vol. 4051, pp. 703–714, 2006) where only symmetric signatures over the Hadamard basis (special basis of size 1) were obtained. In particular, we give a complete list of Boolean symmetric signatures over bases of size 1. It is an open problem whether signatures over bases of higher dimensions are strictly more powerful. The recent result by Valiant (FOCS 2006, pp. 509–517, 2006) seems to suggest that bases of size 2 might be indeed more powerful than bases of size 1. This result is with regard to a restrictive counting version of #SAT called #Pl-Rtw-Mon-3CNF. It is known that the problem is #P-hard, and its mod 2 version is ⊕P-hard. Yet its mod 7 version is solvable in polynomial time by holographic algorithms. This was ac complished by a suitable symmetric signature over a basis of size 2. We show that the same unexpected holographic algorithm can be realized over a basis of size 1. Furthermore we prove that 7 is the only modulus for which such an “accidental algorithm” exists.

Journal ArticleDOI
TL;DR: In this paper, a multilevel zero correlation zone (ZCZ) sequence is derived for CDMA systems and other applications based on the multi-level Hadamard matrices.
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.

01 Jan 2006
TL;DR: In this paper, the authors proposed an indoor wireless infrared downlink scheme for high-data-rate multiuser connectivity with diffuse channels, which is based on synchronous code-division multiple access with unipolar Hadamard codes.
Abstract: We propose an indoor wireless infrared downlink scheme for high-data-rate multiuser connectivity with diffuse channels. The scheme is based on synchronous code-division multiple access with unipolar Hadamard codes. The orthogonality of unipolar Hadamard codes enables multiuser operation with relatively short codes. Thus, practical downlink rates of tens of Mb/s for each user can be obtained. However, multipath reflec- tions in diffuse channels cause strong multipath dispersion and, consequently, severe distortion. This distortion becomes even more severe in a multiuser environment, as the dispersed incoherent infrared radiation of all users aggregates together. To mitigate this distortion, we use a novel adaptive multilevel serial composite decision feedback and feedforward equalizer. We investigate the system's performance with the proposed equalizer, and compare it with the performance of the same system, both composite decision-feedback and feedforward equalizers, and with a conven- tional decision-feedback equalizer (DFE). Our results show that the proposed scheme enables a high-data-rate multiaccess link and eliminates most of the multiuser distortion. Furthermore, it improves system performance in a multiaccess environment, as compared with the other composite equalizers and DFE for the same complexity. We also compare other coding schemes, and show that Hadamard codes are on top of the other codes. Index Terms—Decision-feedback equalizer (DFE), decision feed- forward equalizer, indoor wireless communication, infrared (IR) communication, multiaccess communication, optical code-division multiple access (OCDMA).

Patent
03 Aug 2006
TL;DR: A block transform-based digital media codec achieves faster performance by re-mapping components of the digital media data into vectors or parallel units on which many operations of the transforms can be performed on a parallel or single-instruction, multiple data (SIMD) basis as discussed by the authors.
Abstract: A block transform-based digital media codec achieves faster performance by re-mapping components of the digital media data into vectors or parallel units on which many operations of the transforms can be performed on a parallel or single-instruction, multiple data (SIMD) basis. In the case of a one-dimensional lapped biorthogonal transform, the digital media data components are re-mapped into vectors on which butterfly stages of both overlap pre-/post-filter and block transform portions of the lapped transform can be performed on a SIMD basis. In the case of a two-dimensional lapped biorthogonal transform, the digital media data components are re-mapped into vectors on which a Hadamard operator of both overlap pre-/post-filter and block transform can be performed on a SIMD basis.

Journal ArticleDOI
TL;DR: In this article, a decoupling scheme based on orthogonal arrays of strength t has been proposed for 2-local qudit Hamiltonians, where tges2 and dges2 can be refined to have time-slots of equal length.
Abstract: Decoupling schemes are used in quantum information processing to selectively switch off unwanted interactions in a multipartite Hamiltonian. A decoupling scheme consists of a sequence of local unitary operations which are applied to the system's qudits and alternate with the natural time evolution of the Hamiltonian. Several constructions of decoupling schemes have been given in the literature. Here we focus on two such schemes. The first is based on certain triples of submatrices of Hadamard matrices that are closed under pointwise multiplication (see Leung, "Simulation and reversal of n-qubit Hamiltonians using Hadamard matrices," J. Mod. Opt., vol. 49, pp. 1199-1217, 2002), the second uses orthogonal arrays (see Stollsteimer and Mahler, "Suppression of arbitrary internal couplings in a quantum register," Phys. Rev. A., vol. 64, p. 052301, 2001). We show that both methods lead to the same class of decoupling schemes. We extend the first method to 2-local qudit Hamiltonians, where dges2. Furthermore, we extend the second method to t-local qudit Hamiltonians, where tges2 and dges2, by using orthogonal arrays of strength t. We also establish a characterization of orthogonal arrays of strength t by showing that they are equivalent to decoupling schemes for t-local Hamiltonians which have the property that they can be refined to have time-slots of equal length. The methods used to derive efficient decoupling schemes are based on classical error-correcting codes

Journal ArticleDOI
TL;DR: Computer simulation, in vitro and in vivo experiments confirm the theoretical derivation of voxel bleed reduction from ∼17% to below 5% per Hadamard‐encoded direction.
Abstract: The point spread function (PSF) of Hadamard encoding deviates from its ideal profile due to practical (as opposed to intrinsic) reasons. Finite radiofrequency (RF) pulse length and gradient strength cause slice profile imperfections that lead to cross-talk ("voxel bleed") as large as 17% for a 1-KHz bandwidth, 5.12-ms RF pulse under 3 mT/m. This could adversely affect localization and quantification, and consequently clinical usefulness. A simple modification of the Hadamard RF pulse synthesis that exploits its unique ability to encode noncontiguous slices is proposed and shown to markedly improve the PSF. Computer simulation, in vitro and in vivo experiments confirm the theoretical derivation of voxel bleed reduction from approximately 17% to below 5% per Hadamard-encoded direction.

Posted Content
TL;DR: This note investigates the connection between tiling of Abelian groups and constructions of complex Hadamard matrices via a natural tiling construction and finds some necessary conditions to be equivalent to a Dita-type matrix.
Abstract: Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabo, we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue of complex Hadamard matrices of small order.

Journal ArticleDOI
TL;DR: In this article, the Hadamard weighted geometric mean of K1,..., Kn, the operator K, satisfies the following inequalities for positive kernel operators on a Banach function space.
Abstract: Let K1, . . . , Kn be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K1, . . . , Kn, the operator K, satisfies the following inequalities