43 research outputs found
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
Complex Hadamard matrices of order 6: a four-parameter family
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 G together with Karlsson's degenerate family K and
Tao's spectral matrix S form an exhaustive list of complex Hadamard matrices of
order 6. Such a complete characterization might finally lead to the solution of
the famous MUB-6 problem.Comment: 17 pages; Contribution to the workshop "Quantum Physics in higher
dimensional Hilbert Spaces", Traunkirchen, Austria, July 201
A note on the existence of BH(19,6) matrices
In this note we utilize a non-trivial block approach due to M. Petrescu to
exhibit a Butson-type complex Hadamard matrix of order 19, composed of sixth
roots of unity.Comment: 3 pages, preprin
A further look into combinatorial orthogonality
Strongly quadrangular matrices have been introduced in the study of the
combinatorial properties of unitary matrices. It is known that if a (0,
1)-matrix supports a unitary then it is strongly quadrangular. However, the
converse is not necessarily true. In this paper, we fully classify strongly
quadrangular matrices up to degree 5. We prove that the smallest strongly
quadrangular matrices which do not support unitaries have exactly degree 5.
Further, we isolate two submatrices not allowing a (0, 1)-matrix to support
unitaries.Comment: 11 pages, some typos are corrected. To appear in The Electronic
journal of Linear Algebr
Hadamard matrices modulo 5
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 n != 3, 7 (mod 10) or n
!= 6, 11. In particular, this solves the 5-modular version of the Hadamard
conjecture.Comment: 7 pages, submitted to JC
A note on five dimensional kissing arrangements
The kissing number is the maximum number of pairwise
non-overlapping unit spheres each touching a central unit sphere in the
-dimensional Euclidean space. In this note we report on how we discovered a
new, previously unknown arrangement of unit spheres in dimension . Our
arrangement saturates the best known lower bound on , and refutes a
`belief' of Cohn--Jiao--Kumar--Torquato.Comment: Workshop on Orthogonal designs and related Combinatorics, Meiji
University, Toky