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...

Holonomic quantum computation is analyzed from geometrical viewpoint. We develop an optimization scheme in which an arbitrary unitary gate is implemented with a small circle in a complex projective space. Exact solutions for the Hadamard, CNOT and 2-qubit discrete Fourier transformation gates are explicitly constructed.

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...

The superadditivity and monotonicity properties of some functionals associated with convex functions and the Hermite-Hadamard inequality in the general setting of linear spaces are investigated. Applications for norms and convex functions of a real variable are given. Some inequalities for arithmetic, geometric, harmonic, logarithmic and identric means are improved. 1. Introduction For any conv...

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...

The product of two free scalar fields on a manifold is shown to be a well defined operator valued distribution on the GNS Hilbert space of a globally Hadamard product state. Viewed as a new field all n-point distributions exist, giving a new example for a Wightman field on a manifold.

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...