Arithmetic crosscorrelation of pseudorandom binary sequences of coprime periods

TL;DR

This paper studies the arithmetic crosscorrelation of binary sequences, showing it remains constant for coprime periods and providing bounds for specific sequence types, enhancing understanding of their pseudorandom properties.

Contribution

It proves the constancy of arithmetic crosscorrelation for coprime period sequences and establishes upper bounds for Legendre and m-sequences.

Findings

Arithmetic crosscorrelation is constant for coprime period sequences.

Upper bounds are derived for Legendre sequences.

Upper bounds are derived for binary m-sequences.

Abstract

The (classical) crosscorrelation is an important measure of pseudorandomness of two binary sequences for applications in communications. The arithmetic crosscorrelation is another figure of merit introduced by Goresky and Klapper generalizing Mandelbaum's arithmetic autocorrelation. First we observe that the arithmetic crosscorrelation is constant for two binary sequences of coprime periods which complements the analogous result for the classical crosscorrelation. Then we prove upper bounds for the constant arithmetic crosscorrelation of two Legendre sequences of different periods and of two binary $m$-sequences of coprime periods, respectively.