52 research outputs found
Maximum-order Complexity and Correlation Measures
We estimate the maximum-order complexity of a binary sequence in terms of its
correlation measures. Roughly speaking, we show that any sequence with small
correlation measure up to a sufficiently large order cannot have very small
maximum-order complexity
Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
In our previous paper \cite{co1} we have shown that the theory of circulant
matrices allows to recover the result that there exists Mutually Unbiased
Bases in dimension , being an arbitrary prime number. Two orthonormal
bases of are said mutually unbiased if
one has that ( hermitian scalar product in ). In this paper we show that the theory of block-circulant matrices with
circulant blocks allows to show very simply the known result that if
( a prime number, any integer) there exists mutually Unbiased
Bases in . Our result relies heavily on an idea of Klimov, Munoz,
Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil
sums for , which generalizes the fact that in the prime case the
quadratic Gauss sums properties follow from our results
On the ground states of the Bernasconi model
The ground states of the Bernasconi model are binary +1/-1 sequences of
length N with low autocorrelations. We introduce the notion of perfect
sequences, binary sequences with one-valued off-peak correlations of minimum
amount. If they exist, they are ground states. Using results from the
mathematical theory of cyclic difference sets, we specify all values of N for
which perfect sequences do exist and how to construct them. For other values of
N, we investigate almost perfect sequences, i.e. sequences with two-valued
off-peak correlations of minimum amount. Numerical and analytical results
support the conjecture that almost perfect sequences do exist for all values of
N, but that they are not always ground states. We present a construction for
low-energy configurations that works if N is the product of two odd primes.Comment: 12 pages, LaTeX2e; extended content, added references; submitted to
J.Phys.
Optimization Methods for Designing Sequences with Low Autocorrelation Sidelobes
Unimodular sequences with low autocorrelations are desired in many
applications, especially in the area of radar and code-division multiple access
(CDMA). In this paper, we propose a new algorithm to design unimodular
sequences with low integrated sidelobe level (ISL), which is a widely used
measure of the goodness of a sequence's correlation property. The algorithm
falls into the general framework of majorization-minimization (MM) algorithms
and thus shares the monotonic property of such algorithms. In addition, the
algorithm can be implemented via fast Fourier transform (FFT) operations and
thus is computationally efficient. Furthermore, after some modifications the
algorithm can be adapted to incorporate spectral constraints, which makes the
design more flexible. Numerical experiments show that the proposed algorithms
outperform existing algorithms in terms of both the quality of designed
sequences and the computational complexity
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