56 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 Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States
We address the problem of constructing positive operator-valued measures
(POVMs) in finite dimension consisting of operators of rank one which
have an inner product close to uniform. This is motivated by the related
question of constructing symmetric informationally complete POVMs (SIC-POVMs)
for which the inner products are perfectly uniform. However, SIC-POVMs are
notoriously hard to construct and despite some success of constructing them
numerically, there is no analytic construction known. We present two
constructions of approximate versions of SIC-POVMs, where a small deviation
from uniformity of the inner products is allowed. The first construction is
based on selecting vectors from a maximal collection of mutually unbiased bases
and works whenever the dimension of the system is a prime power. The second
construction is based on perturbing the matrix elements of a subset of mutually
unbiased bases.
Moreover, we construct vector systems in \C^n which are almost orthogonal
and which might turn out to be useful for quantum computation. Our
constructions are based on results of analytic number theory.Comment: 29 pages, LaTe
Fractional jumps: complete characterisation and an explicit infinite family
In this paper we provide a complete characterisation of transitive fractional
jumps by showing that they can only arise from transitive projective
automorphisms. Furthermore, we prove that such construction is feasible for
arbitrarily large dimension by exhibiting an infinite class of projectively
primitive polynomials whose companion matrix can be used to define a full orbit
sequence over an affine space
On lattice profile of the elliptic curve linear congruential generators
Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order k to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG
On the lattice structure of pseudorandom numbers generated over arbitrary finite fields
10.1007/s002000100074Applicable Algebra in Engineering, Communications and Computing123265-272AAEC
On the distribution of some new explicit nonlinear congruential pseudorandom numbers
Lecture Notes in Computer Science3486266-27
Lattice structure and linear complexity of nonlinear pseudorandom numbers
10.1007/s002000200105Applicable Algebra in Engineering, Communications and Computing134319-326AAEC
Lattice structure and linear complexity profile of nonlinear pseudorandom number generators
10.1007/s00200-003-0116-6Applicable Algebra in Engineering, Communications and Computing136499-508AAEC
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