广义超度量矩阵的封闭性质

本文研究了广义超度量矩阵的封闭性质.证明了若A为非奇异的广义超度量矩阵,则A与A的转置的Hadamard积仍然是一个广义超度量矩阵,并且它的逆矩阵是一个对角占优的M矩阵.给出了两个广义超度量矩阵Hadamard积封闭的一个充分条件.最后,讨论了广义超度量矩阵的Perron补与和的封闭条件.     

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     

On Circulant Complex Hadamard Matrices

We show that a circulant complex Hadamard matrix of order n is equivalent to a relative difference set in the group C ₄×C _n where the forbidden subgroup is the unique subgroup of order two which is contained in the C ₄ component. We obtain non-existence results for these relative difference sets. Our results are sufficient to prove there are no circulant complex Hadamard matrices for many orders.     

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.     

广义循环矩阵的一个性质

可表为非奇异对角矩阵和循环矩阵乘积的矩阵,我们称其为广义循环矩阵.本文给出了单位矩阵与广义循环矩阵的和矩阵的非奇异的充要条件,得到了这样和矩阵的相对增益阵列的显示表达式.     

A Series of Regular Hadamard Matrices

Let p and 2p−1 be prime powers and p ≡ 3 (mod 4). Then there exists a symmetric design with parameters (4p², 2p² − p, p² − p). Thus there exists a regular Hadamard matrix of order 4p².     

Sharp Integral Inequalities of the Hermite–Hadamard Type

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi and Pecari , and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning L^p estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.     

Symmetric <Emphasis Type="Bold">(4,4)</Emphasis>-Nets and Generalized Hadamard Matrices Over Groups of Order 4

The symmetric class-regular (4,4)-nets having a group of bitranslations G of order four are enumerated up to isomorphism. There are 226 nets with , and 13 nets with . Using a (4,4)-net with full automorphism group of smallest order, the lower bound on the number of pairwise non-isomorphic affine 2-(64,16,5) designs is improved to 21,621,600. The classification of class-regular (4,4)-nets implies the classification of all generalized Hadamard matrices (or difference matrices) of order 16 over a group of order four up to monomial equivalence. The binary linear codes spanned by the incidence matrices of the nets, as well as the quaternary and -codes spanned by the generalized Hadamard matrices are computed and classified. The binary codes include the affine geometry [64,16,16] code spanned by the planes in AG(3,4) and two other inequivalent codes with the same weight distribution.These codes support non-isomorphic affine 2-(64,16,5) designs that have the same 2-rank as the classical affine design in AG(3,4), hence provide counter-examples to Hamadas conjecture. Many of the -codes spanned by generalized Hadamard matrices are self-orthogonal with respect to the Hermitian inner product and yield quantum error-correcting codes, including some codes with optimal parameters.Vladimir D. Tonchev-Research of this author sponsored by the National Security Agency under Grant MDA904-03-1-0088.classification 5B, 51E, 94B     

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,

关于Hadamard乘积矩阵的一些性质的注记

对文 [1 ]的主要结论作了说明 ,给出 Hadamard乘积矩阵有关性质的更一般的结果 .     

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.     

Zero-Term Ranks of Real Matrices and Their Preservers

Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve zero-term rank of the m× nreal matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.     

Hadamard Matrices and Dihedral Groups

Let D _2p be a dihedral group of order 2p, where p is an odd integer. Let ZD _2p be the group ring of D _2p over the ring Z of integers. We identify elements of ZD _2p and their matrices of the regular representation of ZD _2p . Recently we characterized the Hadamard matrices of order 28 ([6] and [7]). There are exactly 487 Hadamard matrices of order 28, up to equivalence. In these matrices there exist matrices with some interesting properties. That is, these are constructed by elements of ZD ₆. We discuss relation of ZD _2p and Hadamard matrices of order n=8p+4, and give some examples of Hadamard matrices constructed by dihedral groups.     

Hadamard Matrices Modulo 5

In this paper, we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5‐modular Hadamard matrices of order n if and only if or . In particular, this solves the 5‐modular version of the Hadamard conjecture.     

Hadamard and Conference Matrices

We discuss new constructions of Hadamard and conference matrices using relative difference sets. We present the first example of a relative -difference set where n – 1 is not a prime power.     

广义Nekrasov矩阵的充分条件

对任意给定的矩阵,从矩阵元素出发,利用定义构造特殊的正对角矩阵,结合不等式的放缩和数学归纳法,给出广义Nekrasov矩阵判定的三个充分条件,并用数值实例说明了所得结果的有效性.     

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.     

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 .     

Some New Orders of Hadamard and Skew‐Hadamard Matrices

We construct Hadamard matrices of orders and , and skew‐Hadamard matrices of orders and . As far as we know, such matrices have not been constructed previously. The constructions use the Goethals–Seidel array, suitable supplementary difference sets on a cyclic group and a new efficient matching algorithm based on hashing techniques.