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We study the connection between the correlation decay property (more precisely, strong spatial mixing) and the zero-freeness of the partition function of 2-spin systems on graphs of bounded degree. We show that for 2-spin systems on an entire family of graphs of a given bounded degree, the contraction property that ensures correlation decay exists for certain real parameters implies the zero-freeness of the partition function and the existence of correlation decay for some corresponding complex neighborhoods. Based on this connection, we are able to extend any real parameter of which the 2-spin system on graphs of bounded degree exhibits correlation decay to its complex neighborhood where the partition function is zero-free and correlation decay still exists. We give new zero-free regions in which the edge interaction parameters and the uniform external field are all complex-valued, and we show the existence of correlation decay for such complex regions. As a consequence, we obtain approximation algorithms for computing the partition function of 2-spin systems on graphs of bounded degree for these complex parameter settings.

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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 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.cryptosystems [9,10,27]. The research of new sequences with good correlation properties has been an interesting research issue for decades, especially sequences with ideal two-level autocorrelation [10,18].There has been significant progress in finding new sequences with ideal two-level autocorrelation in the last two decades. In 1997, by exhaustive search, Gong, Gaal and Golomb found a class of binary sequences of period 2 n − 1 with 2-level autocorrelation in [12], and in 1998, No, Golomb, Gong, Lee, and Gaal published five conjectures regarding binary sequences of period 2 n − 1 with ideal two-level autocorrelation [26] including two classes, called Welch-Gong transformation sequences, conjectured by the group of the authors in [12]. Interestingly, using monomial hyperovals, Maschietti constructed three classes of binary sequences of period 2 n − 1 with ideal two-level autocorrelation [24] from Segre and Green type monomial hyper ovals and a shorter proof of those sequences is reported in [28] [4]. Shortly after that, No, Chung, and Yun [25], in terms of the image set of the polynomial z d + (z + 1) d where d = 2 2k − 2 k + 1 where 3k ≡ 1 mod n, a special Kasami exponent, conjectured another class of binary sequences of period 2 n − 1 with ideal two-level autocorrelation. This class turned out to be the same class as the Welch-Gong sequences conjectured in [26] and Dobbertin formally proved that in [7]. In 1999, for the case of n odd, Dillon proved the conjecture of Welch-Gong sequence using the Hadamard transform [4], i.e., he showed that the Welch-Gong sequence is equivalent to an m-sequence under the Hadamard transform. A few months later, Dillon and Dobbertin confirmed all these conjectured classes of ideal two-level autocorrelation sequences of period 2 n − 1, although the paper is published later [6].The progress on binary 2-level autocorrelation sequences has been collected in [10] and has no new sequences coming out since then.The progress on searching for nonbinary sequences with 2-level autocorrelation seems different. For p = 3, Lin conjectured a class of ideal two-level autocorrelation sequences of period 3 n − 1 with two trace monomial terms in 1998 in his Ph.D thesis [22]. In 2001, a new class of ternary idea...