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.