On the proof of Lin's conjecture

Abstract

Abstract—In 1998, Lin presented a conjecture on a classof ternary sequences with ideal 2-level autocorrelation. Thosesequences have a very simple structure, i.e., their trace represen-tation has two trace monomial terms. In this paper, we presenta proof for this conjecture. The mathematical tools employedare the second-order multiplexing decimation-Hadamard trans-form, Stickelberger’s theorem, the Teichmuller character, and¨combinatorial techniques for enumerating the Hamming weightsof ternary numbers. As a by-product, we also prove that theLin conjectured ternary sequences are Hadamard equivalent toternary m-sequences. I. I NTRODUCTION Sequences with good random properties have wide appli-cations in modern communications and cryptography, such asCDMA communication systems, global positioning systems,radar, and stream cipher cryptosystems [9], [10], [28]. Theresearch of new sequences with good correlation propertieshas been an interesting research issue for decades, especiallysequences with ideal two-level autocorrelation [10], [18].There has been significant progress in finding new se-quences with ideal two-level autocorrelation in the last twodecades. In 1997, by exhaustive search, Gong, Gaal andGolomb found a class of binary sequences of period 2