Divisible designs and semi-regular relative difference sets from additive Hadamard cocycles

Additive Hadamard cocycles are a natural generalization of presemifields. In this paper, we study divisible designs and semi-regular relative difference sets obtained from additive Hadamard cocycles. We show that the designs obtained from additive Hadamard cocycles are flag transitive. We introduce a new product construction of Hadamard cocycles. We also study additive Hadamard cocycles whose divisible designs admit a polarity in which all points are absolute. Our main results include generalizations of a theorem of Albert and a theorem of Hiramine from presemifields to additive Hadamard cocycles. At the end, we generalize Maiorana-McFarland?s construction of bent functions to additive Hadamard cocycles.     

亚正定矩阵的Kronecker积

研究亚正定矩阵kronecker积的亚正定性,得到了一个充要条件,同时得到Hadamard积亚正定性的一个充要条件.     

Weaving hadamard matrices with maximum excess and classes with small excess

Weaving is a matrix construction developed in 1990 for the purpose of obtaining new weighing matrices. Hadamard matrices obtained by weaving have the same orders as those obtained using the Kronecker product, but weaving affords greater control over the internal structure of matrices constructed, leading to many new Hadamard equivalence classes among these known orders. It is known that different classes of Hadamard matrices may have different maximum excess. We explain why those classes with smaller excess may be of interest, apply the method of weaving to explore this question, and obtain constructions for new Hadamard matrices with maximum excess in their respective classes. With this method, we are also able to construct Hadamard matrices of near‐maximal excess with ease, in orders too large for other by‐hand constructions to be of much value. We obtain new lower bounds for the maximum excess among Hadamard matrices in some orders by constructing candidates for the largest excess. For example, we construct a Hadamard matrix with excess 1408 in order 128, larger than all previously known values. We obtain classes of Hadamard matrices of order 96 with maximum excess 912 and 920, which demonstrates that the maximum excess for classes of that order may assume at least three different values. Since the excess of a woven Hadamard matrix is determined by the row sums of the matrices used to weave it, we also investigate the properties of row sums of Hadamard matrices and give lists of them in small orders. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 233–255, 2004.     

Direct sums of balanced functions,perfect nonlinear functions,and orthogonal cocycles

Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non‐cyclic abelian groups and use it to find all the orthogonal cocycles over Z ₂^t, 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z ₂^t, t ≥ 2, must be multiplicative. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 173–181, 2008     

Some Observations on Quasi-3 Designs and Hadamard Matrices

We consider Hadamard matrices with the quasi-3 property, and their implications for the existence of certain quasi-symmetric designs. We find necessary conditions on the parameters, and show that the quasi-3 property is preserved by the Kronecker product.     

A system of equations for describing cocyclic Hadamard matrices

Given a basis for 2‐cocycles over a group G of order , we describe a nonlinear system of 4t‐1 equations and k indeterminates over , whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that ( ) is a solution of the system if and only if the 2‐cocycle gives rise to a cocyclic Hadamard matrix . Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups and . A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups , we have found that it suffices to check t instead of the 4t rows of , to decide the Hadamard character of the matrix (for a special class of cocycles f). © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 276–290, 2008     

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.     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

Gold codes,Hadamard partitions and the security of CDMA systems

Let V = {1, 2, . . . , M} and let be a set of Hadamard matrices with the property that the magnitude of the dot product of any two rows of distinct matrices is bounded above. A Hadamard partition is any partition of the set of all rows of the matrices H _i into Hadamard matrices. Such partitions have an application to the security of quasi-synchronous code-division multiple-access radio systems when loosely synchronized (LS) codes are used as spreading codes. A new generation of LS code can be used for each information bit to be spread. For each generation, a Hadamard matrix from some partition is selected for use in the code construction. This code evolution increases security against eavesdropping and jamming. One security aspect requires that the number of Hadamard partitions be large. Thus the number of partitions is studied here. If a Kerdock code construction is used for the set of matrices, the Hadamard partition constructed is shown to be unique. It is also shown here that this is not the case if a Gold (or Gold-like) code construction is used. In this case the number of Hadamard partitions can be enumerated, and is very large.      

Twisted partial actions of Hopf algebras

In this work, the notion of a twisted partial Hopf action is introduced as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras. The conditions on partial cocycles are established in order to construct partial crossed products, which are also related to partially cleft extensions of algebras. Examples are elaborated using algebraic groups.     

2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order,and their related Hadamard matrices and codes

We present the full classification of Hadamard 2-(31,15,7), Hadamard 2-(35, 17,8) and Menon 2-(36,15,6) designs with automorphisms of odd prime order. We also give partial classifications of such designs with automorphisms of order 2. These classifications lead to related Hadamard matrices and self-dual codes. We found 76166 Hadamard matrices of order 32 and 38332 Hadamard matrices of order 36, arising from the classified designs. Remarkably, all constructed Hadamard matrices of order 36 are Hadamard equivalent to a regular Hadamard matrix. From our constructed designs, we obtained 37352 doubly-even [72,36,12] codes, which are the best known self-dual codes of this length until now.      

On -Hadamard matrices

In this paper, we use the Sylvester's approach to construct another Hadamard matrix, namely a J_m-Hadamard matrix, from a given one. Consequently, we can generate other 2^m-1 Hadamard matrices from the constructed J_m-Hadamard matrix. Finally, we also discuss the Kronecker product of an Hadamard matrix and a J_m-Hadamard matrix.     

Rearrangement inequalities for Hermitian matrices

Hermitian matrices can be thought of as generalizations of real numbers. Many matrix inequalities, especially for Hermitian matrices, are derived from their scalar counterparts. In this paper, the Hardy-Littlewood-Pólya rearrangement inequality is extended to Hermitian matrices with respect to determinant, trace, Kronecker product, and Hadamard product.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

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.     

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.     

Subordination Results for p-Valent Meromorphic Functions Associated with a Linear Operator

In this paper, by making use of the Hadamard products, we obtain some subordination results for certain family of meromorphic functions defined by using a new linear operator.     

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     

Hermitian阵的 Hadamard积的Schur补的一些不等式（英文）

本文得到了正定Hermitian阵的Hadamard积的Schur补的一些不等式，进而，给出了他们的一些应用，这些改进了近期的一些结束．     

Subordination Results for ρ-Valent Meromorphic Functions Associated with a Linear Operator

In this paper, by making use of the Hadamard products, we obtain some subordination results for certain family of meromorphic functions defined by using a new linear operator.