The Proof of Lin's Conjecture via the Decimation-Hadamard Transform

TL;DR

This paper proves Lin's conjecture on a class of ternary sequences with ideal autocorrelation using advanced mathematical tools, and shows their equivalence to ternary m-sequences.

Contribution

It provides a rigorous proof of Lin's conjecture and establishes the sequences' Hadamard equivalence to ternary m-sequences, advancing sequence design theory.

Findings

Proof of Lin's conjecture confirmed.

Sequences are Hadamard equivalent to ternary m-sequences.

Utilized advanced algebraic and combinatorial techniques.

Abstract

In 1998, Lin presented a conjecture on a class of ternary sequences with ideal 2-level autocorrelation in his Ph.D thesis. Those sequences have a very simple structure, i.e., their trace representation has two trace monomial terms. In this paper, we present a proof for the conjecture. The mathematical tools employed are the second-order multiplexing decimation-Hadamard transform, Stickelberger's theorem, the Teichm\"{u}ller character, and combinatorial techniques for enumerating the Hamming weights of ternary numbers. As a by-product, we also prove that the Lin conjectured ternary sequences are Hadamard equivalent to ternary $m$-sequences.