Constructions for orthogonal designs using signed group orthogonal designs

Craigen introduced and studied signed group Hadamard matrices extensively and eventually provided an asymptotic existence result for Hadamard matrices. Following his lead, Ghaderpour introduced signed group orthogonal designs and showed an asymptotic existence result for orthogonal designs and consequently Hadamard matrices. In this paper, we construct some interesting families of orthogonal designs using signed group orthogonal designs to show the capability of signed group orthogonal designs in generation of different types of orthogonal designs.Comment: To appear in Discrete Mathematics (Elsevier). No figure

Free nilpotent and HH-type Lie algebras. Combinatorial and orthogonal designs

The aim of our paper is to construct pseudo $H$-type algebras from the covering free nilpotent two-step Lie algebra as the quotient algebra by an ideal. We propose an explicit algorithm of construction of such an ideal by making use of a non-degenerate scalar product. Moreover, as a bypass result, we recover the existence of a rational structure on pseudo $H$-type algebras, which implies the existence of lattices on the corresponding pseudo $H$-type Lie groups. Our approach substantially uses combinatorics and reveals the interplay of pseudo $H$-type algebras with combinatorial and orthogonal designs. One of the key tools is the family of Hurwitz-Radon orthogonal matrices

Asymptotic existence of orthogonal designs

v, 115 leaves ; 29 cmAn orthogonal design of order n and type (si,..., se), denoted OD(n; si,..., se), is a square matrix X of order n with entries from {0, ±x1,..., ±xe}, where the Xj’s are commuting variables, that satisfies XX* = ^ ^g=1 sjx^j In, where X* denotes the transpose of X, and In is the identity matrix of order n. An asymptotic existence of orthogonal designs is shown. More precisely, for any Atuple (s1,..., se) of positive integers, there exists an integer N = N(s1,..., se) such that for each n > N, there is an OD(2n(s1 + ... + se); 2ns1,..., 2nse). This result of Chapter 5 complements a result of Peter Eades et al. which in turn implies that if the positive integers s1, s2,..., se are all highly divisible by 2, then there is a full orthogonal design of type (s1, s2,..., se). Some new classes of orthogonal designs related to weighing matrices are obtained in Chapter 3. In Chapter 4, we deal with product designs and amicable orthogonal designs, and a construction method is presented. Signed group orthogonal designs, a natural extension of orthogonal designs, are introduced in Chapter 6. Furthermore, an asymptotic existence of signed group orthogonal designs is obtained and applied to show the asymptotic existence of orthogonal designs

Glosarium Matematika

273 p.; 24 cm

1 The Amicable-Kronecker Construction of Quaternion Orthogonal Designs

Recently, quaternion orthogonal designs (QODs) were introduced as a mathematical construct with the potential for applications in wireless communications. The potential applications require new methods for constructing QODs, as most of the known methods of construction do not produce QODs with the exact properties required for implementation in wireless systems. This paper uses real amicable orthogonal designs and the Kronecker product to construct new families of QODs. The proposed Amicable-Kronecker Construction can be applied to build quaternion orthogonal designs of a variety of sizes and types. Although it has not yet been simulated whether the resulting designs are useful for applications, their properties look promising for the desired implementations. Furthermore, the construction itself is interesting because it uses a simple family of real amicable orthogonal designs and the Kronecker product as building blocks, opening the door for future construction algorithms using other families of amicable designs and other matrix products. EDICS: SPC-STCD Key words: complex orthogonal designs, real amicable orthogonal designs, quaternions, quaternio

Universe in a glass of iced-water. Exploration in off-the-wall physics

Various exploration in astrophysics has revealed many breakthroughs nowadays, not only with respect to James Webb Telescope, but also recent finding related to water and ice deposits in the Moon surface. Those new findings seem to bring us to new questions related to origin of Earth, Moon and the entire Universe

Universe in a glass of iced-water. Exploration in off-the-wall physics

Various exploration in astrophysics has revealed many breakthroughs nowadays, not only with respect to James Webb Telescope, but also recent finding related to water and ice deposits in the Moon surface. Those new findings seem to bring us to new questions related to origin of Earth, Moon and the entire Universe

Orthogonal designs and complementary sequences: constructions and applications for wireless communication

Problems concerned with the algebraic structure and existence of orthogonal designs and complementary sequences with zero aperiodic autocorrelation have long been of interest in combinatorics, coding theory and applied statistics. Over the last few decades, due to an increased demand for wireless services, new technologies have had to be considered since traditional methods are reaching their technical or financial limits. This has motivated the study of new applications of algebraic structures for wireless communications to improve system performance. For example, space-time block codes (STB Cs) from orthogonal designs for multiple-input multiple-output (M IM O) wireless systems have received a lot of attention due to their inherent orthogonality, which guarantees a full transmit diversity and a simple linear decoding. The objective of this research is to investigate the construction of orthogonal designs and complementary sequences in terms of wireless communication applications. One thread of this thesis is the introduction of amicable orthogonal designs over the real and quaternion domain in the context of STB Cs from orthogonal designs. Another part is the study of Golay complementary sequences and their applications as orthogonal spreading sequences for direct sequence code division multiple access (D S-CD M A) systems. We address the following problems in this thesis: ² The current complex orthogonal STB Cs for more than two antennas are not delay optimal or full rate, and even contain many z eros, which will impede their practical implementation. We first introduce some new complex orthogonal codes of order eight with fewer wasted time slots compared to conventional codes. Furthermore, for those complex STB Cs constructed from amicable orthogonal designs containing z eros and irrational numbers in their coefficient matrices, we use the representation theory of Cliord algebras to improve the form of complex orthogonal STB Cs. B y applying this proposed method, we can construct square, maximum rate complex codes with less z eros and no irrational numbers, e.g. transmitted symbols are equally dispersed through transmit antennas. ² However, given any arbitrary order and type, many amicable orthogonal designs are left undecided, since they cannot be constructed using current techniques. We thus introduce the concept of orthogonal designs equivalence. By searching all the equivalence classes of orthogonal designs we find new amicable orthogonal designs of order eight. In addition, some undecided cases can be concluded with non-existence after searching all the equivalence classes of an orthogonal design with the same order and type. We then summarize all the existence and non-existence results of amicable orthogonal designs with order eight. ² Motivated by the success of space-time block codes from orthogonal designs, we also discuss the construction of amicable orthogonal designs over the quaternion domain (AOD Q ) for their possible applications as space-time-polarization block codes, since the additional polarization diversity can be modelled by means of quaternions. We construct some new AOD Q s using the Kronecker product with real amicable orthogonal designs or real weighing matrices from an amicable family. ² Complementary pairs and orthogonal spreading sequences constructed on the basis of complementary sequences have long been studied for channel separation in D S-CD M A systems. H owever, it is hard to generate sequences with both good autocorrelation and cross-correlation properties so as to achieve a good interference performance. We derive a new class of quadri-phase orthogonal spreading sequences from small mutually orthogonal (M O) complementary sets. We then apply a sequence modification method based on choosing a diag- onal H -equivalent matrix to optimize the correlation properties of sequences. The modifed sequences exhibit a reasonable compromise between autocorrela- tion and cross-correlation characteristics, and, as shown through simulations, lead to good performance when used in an asynchronous multi-user CD M A system