Low Autocorrelation Binary Sequences

TL;DR

This paper reviews recent algorithms and introduces a new method for finding optimal low autocorrelation binary sequences, achieving improved computational efficiency and solving sequences up to length 66 and 119 for specific cases.

Contribution

A novel algorithm with time complexity Θ(N·1.73^N) for finding optimal sequences, enabling computation of sequences up to length 66 and skew-symmetric sequences up to length 119.

Findings

Successfully computed all optimal sequences for N ≤ 66.

Computed all optimal skew-symmetric sequences for N ≤ 119.

Presented a new algorithm with improved efficiency for sequence optimization.

Abstract

Binary sequences with minimal autocorrelations have applications in communication engineering, mathematics and computer science. In statistical physics they appear as groundstates of the Bernasconi model. Finding these sequences is a notoriously hard problem, that so far can be solved only by exhaustive search. We review recent algorithms and present a new algorithm that finds optimal sequences of length $N$ in time $\Theta(N\,1.73^N)$. We computed all optimal sequences for $N\leq 66$ and all optimal skewsymmetric sequences for $N\leq 119$.