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On the ground states of the Bernasconi model

Stephan Mertens and Christine Bessenrodt

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Abstract

The ground states of the Bernasconi model are binary $\pm1$ 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.

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BiBTeX Entry

@article{,
   author    = {Stephan Mertens and Christine Bessenrodt},
   title     = {On the ground states of the {B}ernasconi model},
   year      = {1998},
   journal   = {J.~Phys.~A},
   pages     = {3731-3749},
   volume    = {31}
}

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mertens-bessenrodt.pdf