Proof of a Conjecture of Helleseth: Maximal Linear Recursive Sequences of Period $2^{2^n}-1$ Never Have Three-Valued Cross-Correlation

TL;DR

This paper proves Helleseth's conjecture that binary maximal linear sequences with periods of the form 2^{2^n}-1 cannot have a cross-correlation function taking exactly three distinct values.

Contribution

The paper provides a proof confirming that such sequences never exhibit three-valued cross-correlation, resolving a long-standing conjecture in sequence theory.

Findings

Maximal linear sequences of period 2^{2^n}-1 do not have three-valued cross-correlation.

The proof confirms the conjecture for all n ≥ 0.

This result impacts sequence design in communications and cryptography.

Abstract

We prove a conjecture of Helleseth that claims that for any $n \geq 0$, a pair of binary maximal linear sequences of period $2^{2^n}-1$ can not have a three-valued cross-correlation function.