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

In 1998, Lin presented a conjecture on a class of ternary sequences with ideal 2-level autocor-relation in his Ph.D thesis. Those sequences have a very simple structure, i.e., their trace repre-sentation 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 Teichmüller 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. Index Terms. Teichmüller character, decimation-Hadamard transform, multiplexing decimation-Hadamard transform, Stickelberger’s theorem, two-level autocorrelation.