614 research outputs found
Exotic complex Hadamard matrices, and their equivalence
In this paper we use a design theoretical approach to construct new,
previously unknown complex Hadamard matrices. Our methods generalize and extend
the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a
theoretical explanation for the existence of some sporadic examples of complex
Hadamard matrices in the existing literature. As it is increasingly difficult
to distinguish inequivalent matrices from each other, we propose a new
invariant, the fingerprint of complex Hadamard matrices. As a side result, we
refute a conjecture of Koukouvinos et al. on (n-8)x(n-8) minors of real
Hadamard matrices.Comment: 10 pages. To appear in Cryptography and Communications: Discrete
Structures, Boolean Functions and Sequence
On quaternary complex Hadamard matrices of small orders
One of the main goals of design theory is to classify, characterize and count
various combinatorial objects with some prescribed properties. In most cases,
however, one quickly encounters a combinatorial explosion and even if the
complete enumeration of the objects is possible, there is no apparent way how
to study them in details, store them efficiently, or generate a particular one
rapidly. In this paper we propose a novel method to deal with these
difficulties, and illustrate it by presenting the classification of quaternary
complex Hadamard matrices up to order 8. The obtained matrices are members of
only a handful of parametric families, and each inequivalent matrix, up to
transposition, can be identified through its fingerprint.Comment: 7 page
Complex Hadamard matrices and Equiangular Tight Frames
In this paper we give a new construction of parametric families of complex
Hadamard matrices of square orders, and connect them to equiangular tight
frames. The results presented here generalize some of the recent ideas of
Bodmann et al. and extend the list of known equiangular tight frames. In
particular, a (36,21) frame coming from a nontrivial cube root signature matrix
is obtained for the first time.Comment: 6 pages, contribution to the 16th ILAS conference, Pisa, 201
How much complementarity?
Bohr placed complementary bases at the mathematical centre point of his view
of quantum mechanics. On the technical side then my question translates into
that of classifying complex Hadamard matrices. Recent work (with Barros e Sa)
shows that the answer depends heavily on the prime number decomposition of the
Hilbert space. By implication so does the geometry of quantum state space.Comment: 6 pages; talk at the Vaxjo conference on Foundations of Probability
and Physics, 201
Determinants of (â1,1)-matrices of the skew-symmetric type: a cocyclic approach
An n by n skew-symmetric type (â1, 1)-matrix K = [ki,j ] has 1âs on the main
diagonal and ±1âs elsewhere with ki,j = âkj,i. The largest possible determinant of such
a matrix K is an interesting problem. The literature is extensive for n 0 mod 4 (skew-
Hadamard matrices), but for n 2 mod 4 there are few results known for this question.
In this paper we approach this problem constructing cocyclic matrices over the dihedral
group of 2t elements, for t odd, which are equivalent to (â1, 1)-matrices of skew type.
Some explicit calculations have been done up to t = 11. To our knowledge, the upper
bounds on the maximal determinant in orders 18 and 22 have been improved.Junta de AndalucĂa FQM-01
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