1,036 research outputs found
On the Monomiality of Nice Error Bases
Unitary error bases generalize the Pauli matrices to higher dimensional
systems. Two basic constructions of unitary error bases are known: An algebraic
construction by Knill, which yields nice error bases, and a combinatorial
construction by Werner, which yields shift-and-multiply bases. An open problem
posed by Schlingemann and Werner (see
http://www.imaph.tu-bs.de/qi/problems/6.html) relates these two constructions
and asks whether each nice error basis is equivalent to a shift-and-multiply
basis. We solve this problem and show that the answer is negative. However, we
also show that it is always possible to find a fairly sparse representation of
a nice error basis.Comment: 6 page
Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18
All generalized Hadamard matrices of order 18 over a group of order 3,
H(6,3), are enumerated in two different ways: once, as class regular symmetric
(6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a
group of order 3 acting semi-regularly on points and blocks, and secondly, as
collections of full weight vectors in quaternary Hermitian self-dual codes of
length 18. The second enumeration is based on the classification of Hermitian
self-dual [18,9] codes over GF(4), completed in this paper. It is shown that up
to monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and
245 inequivalent Hermitian self-dual codes of length 18 over GF(4).Comment: 17 pages. Minor revisio
Cohomology-Developed Matrices -- constructing families of weighing matrices and automorphism actions
The aim of this work is to construct families of weighing matrices via their
automorphism group action. This action is determined from the
-cohomology groups of the underlying abstract group. As a consequence,
some old and new families of weighing matrices are constructed. These include
the Paley Conference, the Projective-Space, the Grassmannian, and the
Flag-Variety weighing matrices. We develop a general theory relying on low
dimensional group-cohomology for constructing automorphism group actions, and
in turn obtain structured matrices that we call \emph{Cohomology-Developed
matrices}. This "Cohomology-Development" generalizes the Cocyclic and Group
Developments. The Algebraic structure of modules of Cohomology-Developed
matrices is discussed, and an orthogonality result is deduced. We also use this
algebraic structure to define the notion of \emph{Quasiproducts}, which is a
generalization of the Kronecker-product
Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10
AbstractAll Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291–303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,pm) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71–87]
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