Characterization of $m$-Sequences of Lengths $2^{2k}-1$ and $2^k-1$ with Three-Valued Crosscorrelation

TL;DR

This paper investigates the crosscorrelation properties of pairs of m-sequences with lengths 2^{2k}-1 and 2^k-1, discovering new three-valued crosscorrelation pairs and providing a complete distribution analysis.

Contribution

It introduces new pairs of m-sequences with three-valued crosscorrelation and determines their correlation distribution, advancing understanding of sequence correlation properties.

Findings

Identified new pairs of m-sequences with three-valued crosscorrelation.

Determined the complete correlation distribution for these sequence pairs.

Conjectured no other three-valued crosscorrelation cases exist beyond those proven.

Abstract

Considered is the distribution of the crosscorrelation between $m$-sequences of length $2^m-1$, where $m=2k$, and $m$-sequences of shorter length $2^k-1$. New pairs of $m$-sequences with three-valued crosscorrelation are found and the complete correlation distribution is determined. Finally, we conjecture that there are no more cases with a three-valued crosscorrelation apart from the ones proven here.