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 ...

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...

We report on a search for mutually unbiased bases (MUBs) in 6 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 ...