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

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

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 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

Preinjective subfactors of preinjective Kronecker modules

Using a representation theoretical approach we give an explicit numerical characterization in terms of Kronecker invariants of the subfactor relation between two preinjective (and dually preprojective) Kronecker modules, describing explicitly a so called linking module as well. Preinjective (preprojective) Kronecker modules correspond to matrix pencils having only minimal indices for columns (respectively for rows). Thus our result gives a solution to the subpencil problem in these cases (including the completion), moreover the required computations are straightforward and can be carried out easily (both for checking the subpencil relation and constructing the completion pencils based on the linking module). We showcase our method by carrying out the computations on an explicit example.Comment: 13 page