Exhaustive search for low autocorrelation binary sequences

TL;DR

This paper presents an efficient exhaustive search algorithm for finding low autocorrelation binary sequences, significantly advancing the ability to identify groundstates in the Bernasconi-model up to length 48.

Contribution

It introduces a novel exhaustive search algorithm with improved runtime, enabling the compilation of a comprehensive table of exact groundstates for sequences up to length 48.

Findings

Algorithm with runtime $O(1.85^N)$ for exhaustive search

Compiled a table of exact groundstates up to N=48

Suggests the merit factor exceeds 9 as N approaches infinity

Abstract

Binary sequences with low autocorrelations are important in communication engineering and in statistical mechanics as groundstates of the Bernasconi-model. Computer searches are the main tool to construct such sequences. Due to the exponential size $O(2^N)$ of the configuration space, exhaustive searches are limited to short sequences. We discuss an exhaustive search algorithm with run time characteristic $O(1.85^N)$ and apply it to compile a table of exact groundstates of the Bernasconi-model up to $N=48$. The data suggests $F>9$ for the optimal merit factor in the limit $N\to\infty$.