Genetic Algorithm for Orthogonal Designs

We show how to use Simple Genetic Algorithm to produce Hadamard matrices of large orders, from teh full orthogonal design or oder 16 with 9 variables, OD(16; 1, 1, 2, 2, 2, 2, 2, 2, 2). The objective functionthat we use in our implementation of Simple Genetic Algorithm, comes from a Computational Algebra formalism of the full orthogonal design equations. In particular, we constructed Hadamard matrices of orders 144, 176, 208, 240, 272, 304 and 336, from the aforementioned orthogonal design. By varying three genetic operator parameters, we computer 62 inequivalent Hadamard matices of order 304 and 4 inequivalent Hadamard matrices of order 336. Therefore we established two new constructive lower bounds for the numbers of Hadamard matrices of order 304 and 336

Classification Results of Hadamard Matrices

In 1893 Hadamard proved that for any n x n matrix A over the complex numbers, with all of its entries of absolute value less than or equal to 1, it necessarily follows that |det(A)| ≤ nn/2 [n raised to the power n divided by two], with equality if and only if the rows of A are mutually orthogonal and the absolute value of each entry is equal to 1 (See [2], [3]). Such matrices are now appropriately identified as Hadamard matrices, which provides an active area of research in both theoretical and applied fields of the sciences. In pure mathematics, Hadamard matrices are of interest due to their intrisic beauty as well as their applications to areas such as combinatorics, information theory, optics, operator algebras and quantum mechanics. In this text we will introduce some fundamental properties of Hadamard matrices as well as provide a proofs of some classification results for real Hadamard matrices

Do We Really Need Both BEKK and DCC? A Tale of Two Covariance Models

Large and very large portfolios of financial assets are routine for many individuals and organizations. The two most widely used models of conditional covariances and correlations are BEKK and DCC. BEKK suffers from the archetypal "curse of dimensionality" whereas DCC does not. This is a misleading interpretation of the suitability of the two models to be used in practice. The primary purposes of the paper are to define targeting as an aid in estimating matrices associated with large numbers of financial assets, analyze the similarities and dissimilarities between BEKK and DCC, both with and without targeting, on the basis of structural derivation, the analytical forms of the sufficient conditions for the existence of moments, and the sufficient conditions for consistency and asymptotic normality, and computational tractability for very large (that is, ultra high) numbers of financial assets, to present a consistent two step estimation method for the DCC model, and to determine whether BEKK or DCC should be preferred in practical applications.Conditional correlations, Conditional covariances, Diagonal models, Forecasting, Generalized models, Hadamard models, Scalar models, Targeting.

On ZZt × ZZ2 2-cocyclic Hadamard matrices

A characterization of ZZt × ZZ22 -cocyclic Hadamard matrices is described, de- pending on the notions of distributions, ingredients and recipes. In particular, these notions lead to the establishment of some bounds on the number and distribution of 2-coboundaries over ZZt × ZZ22 to use and the way in which they have to be combined in order to obtain a ZZt × ZZ22 -cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in [4] is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries defining ZZt × ZZ22 -cocyclic matrices, which preserve orthogonality. We split the set of Hadamard matrices into disjoint orbits, de- fine representatives for them and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way, in terms of diagrams. Let H be the set of cocyclic Hadamard matrices over ZZt × ZZ22 having a sym- metric diagram. We also prove that the set of Williamson type matrices is a subset of H of size |H| t .Junta de Andalucía FQM-01

Complex Hadamard matrices of order 6: a four-parameter family

In this paper we construct a new, previously unknown four-parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various sextic polynomials. We conjecture that the new, generic family G together with Karlsson's degenerate family K and Tao's spectral matrix S form an exhaustive list of complex Hadamard matrices of order 6. Such a complete characterization might finally lead to the solution of the famous MUB-6 problem.Comment: 17 pages; Contribution to the workshop "Quantum Physics in higher dimensional Hilbert Spaces", Traunkirchen, Austria, July 201