On the ground states of the Bernasconi model

The ground states of the Bernasconi model are binary 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 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 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.