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.

Thesis (Ph. D.)--University of Oregon, 2005. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 58-59). Also available for download via the World Wide Web; free to University of Oregon users.

"The present paper comprises a survey of investigations undertaken to determine possible values of n for which a Hadamard matrix of order n may be constructed"--Introduction. / "August, 1960." / Typescript. / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: M. F. Tinsley, Professor Directing Paper. / Includes bibliographical references (leaf 33).

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

by Yuen Wong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1990. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Basic Results --- p.5 / Chapter §2.1 --- General Results --- p.5 / Chapter §2.2 --- Williamson's Method --- p.8 / Chapter Chapter 3 --- Algorithms And Subroutines --- p.15 / Chapter §3.1 --- Introduction --- p.15 / Chapter §3.2 --- Increasing Determinant Method --- p.15 / Chapter §3.3 --- Williamson's Method - Direct Computation --- p.21 / Chapter §3.4 --- Williamson's Method - Increasing Determinant --- p.26 / Chapter Chapter 4 --- Comparisons And Recommendations On Algorithms --- p.32 / Chapter §4.1 --- Introduction --- p.32 / Chapter §4.2 --- Comparisons And Recommendations On IMPROV(N) --- p.32 / Chapter §4.3 --- Comparisons And Recommendations On GENHA(N) --- p.34 / Chapter §4.4 --- Comparisons And Recommendations On VTID(N) --- p.35 / Chapter §4.5 --- Summary --- p.37 / Chapter Chapter 5 --- Applications Of Hadamard Matrices --- p.38 / Chapter §5.1 --- Hadamard Matrices And Balanced Incomplete Block Designs' --- p.38 / Chapter §5.2 --- Hadamard Matrices And Optimal Weighing Designs --- p.43 / Chapter Chapter 6 --- Conclusion --- p.51 / References --- p.52 / Appendices --- p.53

Existence problems (proving that a set is nonempty) abound in mathematics, so we look for generally applicable solutions (such as optimization techniques). To test and improve these techniques, we apply them to the Hadamard Conjecture (proving that Hadamard matrices exist in dimensions divisible by 4), which is a good example to study since Hadamard matrices have interesting applications (communication theory, quantum information theory, experiment design, etc.), are challenging to find, are easily distinguished from other matrices, are known to exist for many dimensions, etc.. In this thesis we study optimization algorithms (Exhaustive search, Hill Climbing, Metropolis, Gradient methods, generalizations thereof, etc.), improve their performance (when using a Graphical Processing Unit), and use them to attempt to find Hadamard matrices (real, and complex). Finally, we give an algorithm to prove non-trivial lower bounds on the Hamming distance between any given matrix with elements in {+1,-1} and the set of Hadamard matrices, then we use this algorithm to study matrices with similar properties to Hadamard matrices, but which are far away (with respect to the Hamming distance) from them.

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.

Yes / To achieve multiple input multiple output (MIMO) in wireless communication, the orthogonal space-time block coding (OSTBC) is evaluated next. At first, the OSTBC design is extended to include Hadamard matrix, referred to in this work, as traditional Hadamard OSTBC. Next, the Hadamard matrix is imposed on the conventional OSTBC, which is referred to, in this work as, Alamouti-Hadamard STBC. Both the traditional Hadamard OSTBC and the conventional STBC are compared with the Alamouti-Hadamard STBC. It will be shown that imposing the Hadamard conditions over the conventional OSTBC, the performance of the OSTBC 2-transmit antenna scheme can be significantly improved in terms of BER performance. All propositions are well supported with analytical derivations.

Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these matrices, we show that the OD (12;1,1,1,9) is the only orthogonal design constructible from 16 circulant matrices of type OD (4n;1,1,1,4n-3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design constructible from 16 circulant matrices of order 12 on 4 variables is the OD (12;1,1,1,9). It is known that by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To complement our studies we reproduce and important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is the order of a Hadamard matrix and 8n2 - 1 is a prime, there is a productive regular Hadamard matrix of order 16n2(82-1)2. As a corollary, we get many new infinite classes of symmetric designs whenever either of 4n(8n2-1)-1,4n(82-1) +1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work. / iv, 64 leaves : ill., map ; 29 cm.