On the ground states of the Bernasconi model

TL;DR

This paper investigates the ground states of the Bernasconi model, introducing perfect and almost perfect sequences with minimal autocorrelations, and provides constructions and existence results for these sequences based on mathematical theory.

Contribution

It characterizes the existence of perfect sequences using cyclic difference sets and proposes a construction for low-energy configurations when N is a product of two odd primes.

Findings

Perfect sequences exist for specific N values as determined by difference sets.

Almost perfect sequences likely exist for all N, but are not always ground states.

A new construction method for low-energy configurations when N is a product of two odd primes.

Abstract

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.