2,162 research outputs found
Complex spherical codes with two inner products
A finite set in a complex sphere is called a complex spherical -code
if the number of inner products between two distinct vectors in is equal to
. In this paper, we characterize the tight complex spherical -codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric -optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.Comment: 10 pages, to appear in European Journal of Combinatoric
The codes and the lattices of Hadamard matrices
It has been observed by Assmus and Key as a result of the complete
classification of Hadamard matrices of order 24, that the extremality of the
binary code of a Hadamard matrix H of order 24 is equivalent to the extremality
of the ternary code of H^T. In this note, we present two proofs of this fact,
neither of which depends on the classification. One is a consequence of a more
general result on the minimum weight of the dual of the code of a Hadamard
matrix. The other relates the lattices obtained from the binary code and from
the ternary code. Both proofs are presented in greater generality to include
higher orders. In particular, the latter method is also used to show the
equivalence of (i) the extremality of the ternary code, (ii) the extremality of
the Z_4-code, and (iii) the extremality of a lattice obtained from a Hadamard
matrix of order 48.Comment: 16 pages. minor revisio
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
Informational power of the Hoggar SIC-POVM
We compute the informational power for the Hoggar SIC-POVM in dimension 8,
i.e. the classical capacity of a quantum-classical channel generated by this
measurement. We show that the states constituting a maximally informative
ensemble form a twin Hoggar SIC-POVM being the image of the original one under
a conjugation.Comment: 6 double column page
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