‎this paper presents two main results that the singular values of the hadamard product of normal matrices $a_i$ are weakly log-majorized by the singular values of the hadamard product of $|a_{i}|$ and the singular values of the sum of normal matrices $a_i$ are weakly log-majorized by the singular values of the sum of $|a_{i}|$‎. ‎some applications to these inequalities are also given‎. ‎in addi...

ABSTRACT: In this paper, a complete construction of quaternary complex Hadamard matrices of order 2 n is obtained using the method of Sylvester construction and Williamson construction. Williamson construction has been generalized to obtain any kind of Hadamard matrices (Complex or Real Numbers). Non-equivalent family of Hadamard Matrices can be obtained using the Generalized Williamson constru...

Abstract. In this paper we define and verify a subclass of harmonic univalent functions involving the argument of complex-value functions of the form f = h + ¯g and investigate some properties of this subclass e.g. necessary and sufficient coefficient bounds, extreme points, distortion bounds and Hadamard product.Abstract. In this paper we define and verify a subclass of harmonic univalent func...

A Z2Z4-linear Hadamard code of length α + 2β = 2 t is a binary Hadamard code which is the Gray map image of a Z2Z4-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly b t−1 2 c and b t 2c nonequivalent Z2Z4-linear Hadamard codes of length 2t, with α = 0 and α 6= 0, respectively, for all t ≥ 3. In this paper, it is shown that each Z2Z4-linear ...

All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of inequivalent Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard matrices of one type and 26,369 such matrices of another t...

A complex Hadamard matrix, C, of order n has elements 1, -1, i, i and satisfies CC* = nIn where C* denotes the conjugate transpose of C. Let C = [cij] be a complex Hadamard matrix of order n. S(C) = ∑ cij is called the sum of C. 0(C) = │S(C)│ is called the excess of C. We study the excess of complex Hadamard matrices. As an application many real Hadamard matrices of large and maximal excess are...

A Z2Z4-linear Hadamard code of length α+2β = 2 is a binary Hadamard code which is the Gray map image of a Z2Z4-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly b t−1 2 c and b t 2 c nonequivalent Z2Z4-linear Hadamard codes of length 2, with α = 0 and α 6= 0, respectively, for all t ≥ 3. In this paper, it is shown that each Z2Z4-linear Hada...

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c 6= 1 to obtain another complex Hadamard matrix. We give a characterisation of rectang...

Equivalence operations for n-dimensional proper Hadamard matrices are defined. Their equivalence classes are investigated and bounds on the number of such equivalence classes are found to be associated with the number of equivalence classes of 2-dimensional Hadamard matrices. Properties of planes (2-dimensional sections) in an n-dimensional proper Hadamard matrix are investigated. It is shown t...

In this letter, a method to construct good binary and quaternary error correcting codes, called complex Hadamard codes, based on a complex Hadamard matrix is presented. The related properties of the codes are analyzed. In addition, through the operation in Z4 domain, a new simplex soft-decision decoding algorithm for the complex Hadamard codes is also proposed. key words: complex Hadamard matri...