On the ground states of the Bernasconi model

Details On the ground states of the Bernasconi model On the ground states of the Bernasconi model

1997-07-10

arxiv.org/abs/cond-mat/9707104v2

Physics

Condensed Matter

Disordered Systems and Neural Networks

12 pages, LaTeX2e; extended content, added references; submitted to J.Phys.A

Scientific paper

10.1088/0305-4470/31/16/004

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.

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