On an inequivalence criterion for cocyclic Hadamard matrices

Given two Hadamard matrices of the same order, it can be quite difficult to decide whether or not they are equivalent. There are some criteria to determine Hadamard inequivalence. Among them, one of the most commonly used is the 4-profile criterion. In this paper, a reformulation of this criterion in the cocyclic framework is given. The improvements obtained in the computation of the 4-profile of a cocyclic Hadamard matrix are indicated.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298

On the Asymptotic Existence of Hadamard Matrices

It is conjectured that Hadamard matrices exist for all orders $4t$ ($t>0$). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers $k$, there is a Hadamard matrix of order $k2^{[a+b\log_2k]}$, where $a$ and $b$ are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take $a=2$ and $b=0$. Since Seberry's ground-breaking result, which showed that we may take $a=0$ and $b=2$, there have been several improvements where $b$ has been by stages reduced to 3/8. In this paper, we show that for all $\epsilon>0$, the set of odd numbers $k$ for which there is a Hadamard matrix of order $k2^{2+[\epsilon\log_2k]}$ has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.Comment: Keywords: Hadamard matrices, Asymptotic existence, Cocyclic Hadamard matrices, Relative difference sets, Riesel numbers, Extended Riemann hypothesis. (Received 2 August 2008, Available online 18 March 2009

On quasi-orthogonal cocycles

We introduce the notion of quasi-orthogonal cocycle. This is motivated in part by the maximal determinant problem for square {±1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi-Hadamard groups, relative quasi-difference sets, and certain partially balanced incomplete block designs, are proved.Junta de Andalucía FQM-01

Generalized binary arrays from quasi-orthogonal cocycles

Generalized perfect binary arrays (GPBAs) were used by Jedwab to construct perfect binary arrays. A non-trivial GPBA can exist only if its energy is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays (GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a procedure to construct GOBAs based on a characterization of the arrays in terms of 2-cocycles. As a further application, we determine negaperiodic Golay pairs arising from generalized optimal binary sequences of small length.Junta de Andalucía FQM-01

Quasi-Hadamard Full Propelinear Codes

In this paper, we give a characterization of quasi-Hadamard groups in terms of propelinear codes. We define a new class of codes that we call quasi-Hadamard full propelinear codes. Some structural properties of these codes are studied and examples are provided.Junta de Andalucía FQM-016Ministerio de Economía y Competitividad TIN2016-77918-

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

Cohomology-Developed Matrices -- constructing families of weighing matrices and automorphism actions

The aim of this work is to construct families of weighing matrices via their automorphism group action. This action is determined from the $0,1,2$-cohomology groups of the underlying abstract group. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley Conference, the Projective-Space, the Grassmannian, and the Flag-Variety weighing matrices. We develop a general theory relying on low dimensional group-cohomology for constructing automorphism group actions, and in turn obtain structured matrices that we call \emph{Cohomology-Developed matrices}. This "Cohomology-Development" generalizes the Cocyclic and Group Developments. The Algebraic structure of modules of Cohomology-Developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of \emph{Quasiproducts}, which is a generalization of the Kronecker-product

On permanents of Sylvester Hadamard matrices

It is well-known that the evaluation of the permanent of an arbitrary $(-1,1)$-matrix is a formidable problem. Ryser's formula is one of the fastest known general algorithms for computing permanents. In this paper, Ryser's formula has been rewritten for the special case of Sylvester Hadamard matrices by using their cocyclic construction. The rewritten formula presents an important reduction in the number of sets of $r$ distinct rows of the matrix to be considered. However, the algorithm needs a preprocessing part which remains time-consuming in general