Low Correlation Sequences from Linear Combinations of Characters

TL;DR

This paper introduces a method for constructing binary sequence pairs with significantly lower autocorrelation and crosscorrelation demerit factors than random sequences, using linear combinations of multiplicative characters of finite fields.

Contribution

The authors provide explicit constructions and asymptotic formulas for sequence pairs with superior correlation properties, outperforming traditional sequences like m-sequences and Legendre sequences.

Findings

Sequence pairs achieve lower correlation demerit factors than random sequences.

Asymptotic formulas accurately predict correlation performance for modest sequence lengths.

Constructed sequences outperform classical sequences in autocorrelation and crosscorrelation metrics.

Abstract

Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared to random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of $1$. Our constructions provide sequence pairs with crosscorrelation demerit factor significantly less than $1$, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than $1$ (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this study, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.