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In this paper Butson-type complex Hadamard matrices BH(n, q) of order n over the complex qth roots of unity are classified for small parameters by computer-aided methods. The results include a classification of BH(21, 3), BH(16, 4), and BH(14, 6) matrices. There are exactly 72, 1786763, and 167776 such matrices, respectively, up to monomial equivalence. Additionally, an example of a BH(14, 10) matrix is shown for the first time, and the nonexistence of BH(8, 15), BH(11, q) for q ∈ {10, 14, 15}, and BH(13, 15) matrices is proved. [ABSTRACT FROM AUTHOR]

In this paper, we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if [ABSTRACT FROM AUTHOR]

Abstract: A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10×10 and 12×12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14×14 matrix that there cannot be a general method for introducing parameters into these types of matrices. [Copyright &y& Elsevier]

In this paper, we construct a new, previously unknown four-parameter family of complex Hadamard matrices of order 6, the entries of which are described by algebraic functions of roots of various sextic polynomials. We conjecture that the new, generic family G6(4) together with Karlsson's degenerate family K6(3) and Tao's spectral matrix S(0)6 form an exhaustive list of complex Hadamard matrices of order 6. Such a complete characterization might finally lead to the solution of the problem of mutually unbiased bases in ℂ6. [ABSTRACT FROM PUBLISHER]

Complex Hadamard matrices have received considerable attention in the past few years due to their application in quantum information theory. While a complete characterization currently available [5] is only up to order 5, several new constructions of higher order matrices have appeared recently [4, 12, 2, 7, 11]. In particular, the classification of self-adjoint complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in [2], providing a previously unknown non-affine one-parameter orbit. In this paper we classify all dephased, symmetric complex Hadamard matrices with real diagonal of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix F6 and Diţă's matrix D6. This answers a recent question of Bengtsson et al. [3]. [ABSTRACT FROM AUTHOR]