We demonstrate how to efficiently implement extremely high-dimensional compressive imaging of a bi-photon probability distribution. Our method uses fast-Hadamard-transform Kronecker-based compressive sensing to acquire the joint space distribution. We list, in detail, the operations necessary to enable fast-transform-based matrix-vector operations in the joint space to reconstruct a 16.8 millio...

In this work, we investigate the asymptotic stability of zero solution for Caputo-Hadamard fractional dynamic equations on a time scale. We will make use Krasnoselskii fixed point theorem in weighted Banach space to show new results.

Spin models for link invariants were introduced by Jones [7], and their connection to combinatorics was revealed first in [4]. Jaeger and Nomura [6] constructed nonsymmetric spin models for link invariants from Hadamard matrices, and showed that these models give link invariants which depend nontrivially on link orientation. These models are a modification of the Hadamard model originally const...

In this paper, the fast algorithm to Walsh-Hadamard transform (WHT) is studied from the point of view of the optical computation. Firstly, MSD addition and butterfly calculation in discrete WHT are presented in detail. Then the design and implementation scheme of fast algorithm of WHT is proposed and the software simulations to fast WHT are carried out optically. Meanwhile, the computational co...

The precoding scheme based on real Hadamard transform is an effective and flexible way for reducing the peak-to-average power ratio (PAPR) of OFDM signals. However, the reduction capacity of PAPR is limited for real Hadamard transform method. In this paper, the new scheme based on complex Hadamard transform is proposed. The simulation results indicate that the proposed scheme may obtain a 1.2 d...

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley–Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and...

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 Hadamard code with α = 0 is equivalent to a Z2Z4-linear Hadamard code with α 6= 0, so there are only b t 2c nonequivalent Z2Z4-linear Hadamard codes of length 2t . Moreover, the orders...

In this paper, we generalize the Hadamard transform to the the case of lapped transform A matrix A(z) is a lapped Hadamard transform if it satisfies A ~ ( ~ ' ) A ( ~ ) = a~ for some integer a and all the entries of its coefficient matrices are i1. Many methods have been proposed to construct lapped Hadamard matrices. In this paper, we will study these matrices using the theory of paraunitary f...

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we find so...

We report on a search for mutually unbiased bases MUBs in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a ...