Constructing cocyclic Hadamard matrices of order 4p

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p\le 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1mod4$ is equivalent to a HM with one of five distinct block structures, including Williamson‐type and (transposed) Ito matrices. If $p\equiv 3mod4$, then every CHM of order $4p$ is equivalent to a Williamson‐type or (transposed) Ito matrix.     

Power Hadamard matrices

We introduce power Hadamard matrices, in order to study the structure of (group) generalized Hadamard matrices, Butson (generalized) Hadamard matrices and other related orthogonal matrices, with which they share certain common characteristics. The new objects turn out to be as interesting, and perhaps as useful, as the objects that motivated them.We develop a basic theory of power Hadamard matrices, explore these relationships, and offer some new insights into old results. For example, we show that all 4×4 Butson Hadamard matrices are equivalent to circulant ones, and how to move between equivalence classes.We provide, among other new things, an infinite family of circulant Butson Hadamard matrices that extends a known class to include one of each positive integer order.Dedication: In 1974 Jennifer Seberry (Wallis) introduced what was then a totally new structure, orthogonal designs, in order to study the existence and construction of Hadamard matrices. They have proved their worth for this purpose, and have also become an object of interest for their own sake and in applications (e.g., [H.J.V. Tarok, A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory 45 (1999) 1456-1467. [26]]). Since then many other generalizations of Hadamard matrices have been introduced, including some discussed herein. In the same spirit we introduce a new object showing this kind of promise.Seberry's contributions to this field are not limited to her own work, of which orthogonal designs are but one example—she has mentored many young mathematicians who have expanded her legacy by making their own marks in this field. It is fitting, therefore, that our contribution to this volume is a collaboration between one who has worked in this field for over a decade and an undergraduate student who had just completed his third year of study at the time of the work.     

On the asymptotic existence of cocyclic Hadamard matrices

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order ²10+tq whenever . We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order ²4N−2q.     

Ryser's embedding problem for Hadamard matrices

What is the minimum order of a Hadamard matrix that contains an a by b submatrix of all 1's? Newman showed that where c^? denotes the smallest order greater than or equal to c for which a Hadamard matrix exists. It follows that if 4 divides both a and b, and if the Hadamard conjecture is true, then . We establish the improved bounds for min {a,b} ≥ 2. The Hadamard conjecture therefore implies that if 4 divides both 2ab and ?a/2? ?b/2?, then (a, b) = 2 · max {?a/2?b, ?b/2?a}. Our lower bound comes from a counting argument, while our upper bound follows from a sub‐multiplicative property of : Improvements in our upper bound occur when suitable conference matrices or Bush‐type Hadamard matrices exist. We conjecture that any (1,?1)‐matrix of size a by b occurs as a submatrix of some Hadamard matrix of order at most . © 2005 Wiley Periodicals, Inc. J Combin Designs     

Hadamard matrices with few distinct types

The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. In this paper, we investigate Hadamard matrices with few distinct types. Among other results, the Sylvester Hadamard matrices are shown to be characterized by their spectrum of types.     

Multilevel and multidimensional Hadamard matrices

Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to {±1}, were introduced by Trinh, Fan, and Gabidulin in 2006 as a way to construct multilevel zero-correlation zone sequences, which have been studied for use in approximately synchronized code division multiple access systems. We answer the open question concerning the maximum number of distinct elements permissible in an order n MHM by proving the existence of an order n MHM with n elements of distinct absolute value for all n. We also define multidimensional MHMs and prove an analogous existence result.      

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.     

On ‐Cocyclic Hadamard Matrices

In this paper, we describe some necessary and sufficient conditions for a set of coboundaries to yield a cocyclic Hadamard matrix over the dihedral group . Using this characterization, new classification results for certain cohomology classes of cocycles over are obtained, extending existing exhaustive calculations for cocyclic Hadamard matrices over from order 36 to order 44. We also define some transformations over coboundaries, which preserve orthogonality of ‐cocycles. These transformations are shown to correspond to Horadam's bundle equivalence operations enriched with duals of cocycles.     

On ‐Cocyclic Hadamard Matrices

A characterization of ‐cocyclic Hadamard matrices is described, depending 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 to use and the way in which they have to be combined in order to obtain a ‐cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in A. Baliga, K. J. Horadam, Australas. J. Combin., 11 (1995), 123–134 is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries defining ‐cocyclic matrices, which preserve orthogonality. We split the set of Hadamard matrices into disjoint orbits, define 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 be the set of cocyclic Hadamard matrices over having a symmetric diagram. We also prove that the set of Williamson‐type matrices is a subset of of size .     

Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs

Suppose there exists a Hadamard 2-$(m,\frac{m?1}{2},\frac{m?3}{4})$ design having skew incidence matrix. If there exists a conference graph on $2m?1$ vertices, then there exists a regular Hadamard matrix of order $4m^2$. A conference graph on $2m+3$ vertices yields a regular Hadamard matrix of order $4{(m+1)}^2$.     

Balancedly splittable Hadamard matrices

Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed.     

Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets

Cocyclic matrices have the form where G is a finite group, C is a finite abelian group and : G × G C is a (two-dimensional) cocycle; that is,

Mappings of Butson-type Hadamard matrices

A $\mathrm{BH}(q,n)$ Butson-type Hadamard matrix $H$ is an $n\times n$ matrix over the complex $q$th roots of unity that fulfils $H{H}^1=n{I}_n$. It is well known that a $\mathrm{BH}(4,n)$ matrix can be used to construct a $\mathrm{BH}(2,2n)$ matrix, that is, a real Hadamard matrix. This method is here generalised to construct a $\mathrm{BH}(q,pn)$ matrix from a $\mathrm{BH}(pq,n)$ matrix, where $q$ has at most two distinct prime divisors, one of them being $p$. Moreover, an algorithm for finding the domain of the mapping from its codomain in the case $p=q=2$ is developed and used to classify the $\mathrm{BH}(4,16)$ matrices from a classification of the $\mathrm{BH}(2,32)$ matrices.     

A sensitive algorithm for detecting the inequivalence of Hadamard matrices

A Hadamard matrix of side is an matrix with every entry either or , which satisfies . Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Hadamard matrices by a complete search is known to be an NP hard problem when increases. In this paper, a new algorithm for detecting inequivalence of two Hadamard matrices is proposed, which is more sensitive than those known in the literature and which has a close relation with several measures of uniformity. As an application, we apply the new algorithm to verify the inequivalence of the known inequivalent Hadamard matrices of order ; furthermore, we show that there are at least pairwise inequivalent Hadamard matrices of order . The latter is a new discovery.

     

Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices

In answer to “Research Problem 16” in Horadam's recent book Hadamard matrices and their applications, we provide a construction for generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 456–458, 2009