We define several operations that switch substructures of Hadamard matrices thereby producing new, generally inequivalent, Hadamard matrices. These operations have application to the enumeration and classification of Hadamard matrices. To illustrate their power, we use them to greatly improve the lower bounds on the number of equivalence classes of Hadamard matrices in orders 32 and 36 to 3,578...

In this paper we study a one-dimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetry of probability distributions for the Hadamard walk.

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...

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...

In this paper, the properties of the WalshHadamard codes used as spreading codes in QCDMA cellular systems are analyzed. It is shown that among Walsh-Hadamard codes, obtained from Hadamard Matrix , only k 2 ) , 2 ( k H k 1 + k codes are orthogonal to each other in all phases. Also, the performance of the Walsh Hadamard codes is compared with that of the Gold codes from the average cross-correla...

The Hadamard conjecture is that Hadamard matrices exist for all orders 1,2, 4t where t 2 1 is an integer. We have obtained the following results which strongly support the conjecture: (i) Given any natural number q, there exists an Hadamard matrix of order 2 q for every s 2 [2log2 (q 3)]. (ii) Given any natural number q, there exists a regular symmetric Hadamard matrix with constant diagonal of...