We make use of a structure known as signed groups, and known sequences with zero autocorrelation to derive new results on the asymptotic existence of Hadamard matrices.

We make use of a structure known as signed groups, and known sequences with zero autocorrelation to derive new results on the asymptotic existence of Hadamard matrices.

Title: Hadamard matrices and their applications in cryptography Author: Jan Luber Department: Department of Algebra Supervisor: prof. RNDr. Aleš Drápal, CSc., DSc., Department of Algebra Abstract: This thesis takes interest in Hadamard matrices, their constructions and application in cryptography. Firstly, we introduce basic properties of Hadamard matrices and selected summary of classical constructions is presented. Then we show a table of constructions that can be used to construct Hadamard matrix of given order. In the next part, we get concerned with Hadamard matrices with circulant cores with detailed description of construction Hadamard matrices with two circulant cores from GL-pair. In the end, we present cryptosystem using Hadamard matrices, we show its essential weaknesses and simple attacks. We propose several improvements in the form of adding other security elements. Keywords: Hadamard matrix, Hadamard conjecture, symmetric cryptography

Yes / In this paper, a simplified approach for implementing QO-STBC is presented. It is based on the Hadamard matrix, in which the scheme exploits the Hadamard property to attain full diversity. Hadamard matrix has the characteristic that diagonalizes a quasi-cyclic matrix and decoding matrix that are diagonal matrix permit linear decoding. Using quasi-cyclic matrices in designing QO-STBC systems require that the codes should be rotated to reasonably separate one code from another such that error floor in the design can be minimized. It will be shown that, orthogonalizing the secondary codes and then imposing the Hadamard criteria that the scheme can be well diagonalized. The results of this simplified approach demonstrate full diversity and better performance than the interference-free QO-STBC. Results show about 4 dB gain with respect to the traditional QO-STBC scheme and performs alike with the earlier Hadamard based QO-STBC designed with rotation. These results achieve the consequent mathematical proposition of the Hadamard matrix and its property also shown in this study.

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In this thesis we introduce selected classes of matrices, whose entries are only numbers −1, 1, 0. We combine existing results from various fields of Mathematics and enrich them with specific examples and explanations, with the aim of making the understanding of the text easier. Thanks to that, the reader can comprehend the theory and look under the hood of non-trivial applications. We will start with introducing adjacency matrices and covering of complete graphs with complete bipartite graphs. Then we follow with Hadamard matrices and will show the conditions for their constructions. Incidence matrices of the set systems will help us solve the combinatorial problem of the Odd-town clubs. Finally, we will prove the Cayley formula about the spanning trees of the complete graph, using incidence matrices.

Hyperspectral imagery is often associated with high storage and transmission costs. Dimensionality reduction aims to reduce the time and space complexity of hyperspectral imagery by projecting data into a low-dimensional space such that all the important information in the data is preserved. Dimensionality-reduction methods based on transforms are widely used and give a data-dependent representation that is unfortunately costly to compute. Recently, there has been a growing interest in data-independent representations for dimensionality reduction; of particular prominence are random projections which are attractive due to their computational efficiency and simplicity of implementation. This dissertation concentrates on exploring the realm of computationally fast and efficient random projections by considering projections based on a random Hadamard matrix. These Hadamard-based projections are offered as an alternative to more widely used random projections based on dense Gaussian matrices. Such Hadamard matrices are then coupled with a fast singular value decomposition in order to implement a two-stage dimensionality reduction that marries the computational benefits of the data-independent random projection to the structure-capturing capability of the data-dependent singular value transform. Finally, random projections are applied in conjunction with nonnegative least squares to provide a computationally lightweight methodology for the well-known spectral-unmixing problem. Overall, it is seen that random projections offer a computationally efficient framework for dimensionality reduction that permits hyperspectral-analysis tasks such as unmixing and classification to be conducted in a lower-dimensional space without sacrificing analysis performance while reducing computational costs significantly.

Yes / A quasi-orthogonal space time block coding (QO-STBC) scheme that exploits Hadamard matrix properties is studied and evaluated. At first, an analytical solution is derived as an extension of some earlier proposed QO-STBC scheme based on Hadamard matrices, called diagonalized Hadamard space-time block coding (DHSBTC). It explores the ability of Hadamard matrices that can translate into amplitude gains for a multi-antenna system, such as the QO-STBC system, to eliminate some off-diagonal (interference) terms that limit the system performance towards full diversity. This property is used in diagonalizing the decoding matrix of the QOSTBC system without such interfering elements. Results obtained quite agree with the analytical solution and also reflect the full diversity advantage of the proposed QO-STBC system design scheme. Secondly, the study is extended over an interference-free QO-STBC multi-antenna scheme, which does not include the interfering terms in the decoding matrix. Then, following the Hadamard matrix property advantages, the gain obtained (for example, in 4x1 QO-STBC scheme) in this study showed 4-times louder amplitude (gain) than the interference-free QOSTBC and much louder than earlier DHSTBC for which the new approach is compared with.

Radio frequency interference is an ever growing problem in the wireless community. This dissertation presents methods to reduce interference for vehicular multi-antenna devices. This document is organized into two parts: the main chapters and the appendices. The main chapters present research conducted primarily by the author. These deserve the reader's primary attention. The appendices showcase contributions made by the author serving in a supporting role to projects led by others and/or do not fit the vehicular theme. These should receive secondary attention. The main chapter contributions are summarized as follows. A device was created that provides over 105 dB of transmit to receive isolation in a full duplex printed circuit board radio. This technology can improve the effective range of vehicular radar systems and increase the bandwidth of full duplex communication schemes for vehicles. The technologies involved are compatible with existing circuit board topologies and are mindful of the size and weight requirements for vehicular use. This isolation performance pushes the state of the art for printed circuit board designs and provides greater capability for these kinds of devices. Recent system on chip computing architectures are opening new pathways for integrating phased array technologies into a single chip. The computer engineering required to configure these devices is beyond the capabilities of many vehicle systems engineers, inviting the author to use one to implement a 16 antenna adaptive beamformer for GPS. The adaptive beamformer can combat multipath bounces and malicious spoofing from ground sources. The high rate analog conversion architecture eliminates the local oscillator distribution to simplify the analog front end to an active antenna. This allows vehicular phased arrays to use smaller footprints and suggests that multi-antenna beamforming devices may be easier to deploy on small to midsized vehicles. Bench tests of the beamformer indicate it can adapt to the environment and increase the received signal strength suggesting it can improve GPS quality for active deployments. The bank of subspace projection beamformers is a popular choice for mitigating interference in digital phased array receivers. A method was discovered that maps that matrix operator into a circuit topology that is simple to implement in an analog circuit and cancels across the entire bandwidth simultaneously. This can offload computational interference mitigation from the signal processor while still allowing secondary multi-pixel digital beamforming downstream. This beamformer was analytically connected to the body of phased array literature and studied to estimate practical error bounds and design methods of calibration.

在本篇論文中我們使用矩陣來建構可分組設計(GDD), 我們列出了兩種型 式的建構, 第一種 -- 起因於 W.H. Haemers -- A .crtimes. J + I .crtimes. D, 利用此種建構我們將所有符合 r - .lambda.1 = 1 的 (m,n,k,.lambda.1,.lambda.2) GDD 分成三類: (i) A=0 或 J-I, (ii) A 為 .mu. - .lambda. = 1 強則圖的鄰接矩陣, (iii) J-2A 為斜對稱 矩陣的核心。第二種型式為 A .crtimes. D + .Abar .crtimes. .Dbar ,此種方法可以建構出 b=4(r-.lambda.2) 的正規和半正規 GDD 。另外在 論文中, 我們研究在這些建構中出現的相關題目。 / In this thesis we use matrices to construct group divisible designs (GDDs). We list two type of constructions, the first type is -- due to W.H. Heamers -- A .crtimes. J + I .crtimes. D and use this construction we classify all the (m,n,k,. lambda.1, .lambda.2) GDD with r - .lambda.1 = 1 in three classes according to (i) A = 0 or J-I, (ii) A is the adjacency matrix of a strongly regular graph with .mu. - .lambda. = 1, (iii) J - 2A is the core of a skew-symmetric Hadamard matrix. The second type is A .crtimes. D + .Abar .crtimes. .Dbar , this type can construct many regular and semi-regular GDDs with b=4(r-.lambda.2). In the thesis we investigate related topics that occur in these constructions.