Non-existence of perfect binary sequences

Abstract: Binary sequences with lower autocorrelation values have important applications in cryptography and communications. In this paper, we present all possible parameters for binary periodical sequences with a 2-level autocorrelation values. For $n \equiv 1\pmod 4$, we prove some cases of Schmidt's Conjecture for perfect binary sequences. (Des. Codes Cryptogr. 78 (2016), 237-267.) For $n \equiv 2\pmod 4$, Jungnickel and Pott (Discrete Appl. Math. 95 (1999) 331-359.) left four perfect binary sequences as open problem and we solve three of its. For $n \equiv 3\pmod 4$, we present some nonexistence of binary sequences which all nontrivial autocorrelation values are equal 3. For $n \equiv 0\pmod 4$, we show that there do not exist the binary sequences which all nontrivial autocorrelation values are equal 4.