New Binary Sequences With Different Periods

Binary sequence is an important research branch of algebraic coding theory. It has important applications in digital computation, digital communications, informa-tion theory, control theory etc. Meanwhile, sequences have a close connection with weight distribution of cyclic code. Cyclic code is a very important linear code and its weight distribution not only determine the error correcting capability, but also is helpful to improve the probability of error detection. We apply linear code such as cyclic code to check or correct information during its transmission, which has become an important way to improve the quality of digital communication transmission. For these reasons, studying sequences and linear code have extremely important theoret-ical and practical significances. In recent years, theory of quadratic form on finite field is applied to the distribution of some cyclic codes. Meanwhile, results on se-quence correlation have also been obtained. In this thesis, we also apply the theory of quadratic form to study the properties of binary m-sequences.Let h,m be two odd integers, gcd(h,m) = 1 and 0 < h < m. In this thesis, we study new pairs of binary sequences. One is m-sequence (ut) of the length 2n - 1.The other is its d-decimated sequence (vt)=(udt)of length 3 (2n - 1). We show that the cross correlation is three-valued. Precisely, -1 occurs (2m+1)(7·2m-8)/9 times;-1+ 2m+1 occurs (2m+1)2/9 times and -1 - 2m+1 occurs (2m+1)(2m-2)/9 times.