New Classes of Binary Sequences with High Merit Factor

TL;DR

This paper introduces new classes of binary sequences with high merit factors, expanding the known sequence types and providing extensive empirical results for sequences of various lengths, which are valuable in multiple scientific fields.

Contribution

The work presents a novel class of binary sequences with even lengths and high merit factors, along with sub-classes based on partition problems and potentials, broadening the scope beyond skew-symmetric sequences.

Findings

Sequences with lengths less than 225 and high MF discovered

Sequences with lengths greater than 225 and high MF identified

High MF sequences for lengths less than 2^8 and 2^9 are revealed

Abstract

The Merit Factor (MF) measure was first introduced by Golay in 1972. Sequences possessing large values of MF are of great interest to a rich list of disciplines - from physics and chemistry to digital communications, signal processing, and cryptography. Throughout the last half-century, manifold approaches and strategies were proposed for finding such sequences. Referenced as one of the most difficult optimization problems, Golay wrote that it is a "challenging and charming problem". His publications on this problem spanned more than 20 years. Golay himself introduced one beneficial class of sequences, called skew-symmetric sequences, or finite binary sequences with odd lengths with alternate autocorrelation values equal to 0. Their sieving construction greatly reduces the computational efforts of finding binary sequences with odd lengths and high MF. Having this in mind, the majority of papers to be found in the literature are focused solely on this class, preferring them over binary sequences with even lengths. In this work, a new class of finite binary sequences with even lengths with alternate autocorrelation absolute values equal to 1 is presented. We show that the MF values of the new class are closely related to the MF values of adjacent classes of skew-symmetric sequences. We further introduce new sub-classes of sequences using the partition number problem and the notion of potentials, measured by helper ternary sequences. Throughout our experiments, MF records for binary sequences with many lengths less than 225, and all lengths greater than 225, are discovered. Binary sequences of all lengths, odd or even, less than $2^8$ and with MF $>8$, and all lengths, odd or even, less than $2^9$ and with MF $>7$, are now revealed.