Cyclotomic Function Field With Module X~2 And Construction Of A Family Of Low Correlation Sequences

The main content of this paper is to construct a family of binary sequences with length of odd prime p based on properties of the cyclotomic function field of modulus x2 over the rational function field Fp(x),and to discuss the auto-and cross-correlations of the sequences constructed based on existing techniques.The specific structure of the article is as follows:The first chapter is a brief introduction,we mainly introduce some existing work on family of sequences with a low correlation,and briefly explains the main results of this paper.In the second chapter,we briefly introduce some basic definitions and related theories about functional field.In the third chapter,we construct a family of binary sequences by using the cyclotomic function field with modulus x2 over the rational function field Fp(x)and discuss their correlation.Firstly,we study the basic properties of the cyclotomic function field with modulus x2;secondly,equivalence classes are divided in the selected RiemannRoch space,and suitable representative elements of equivalence classes are selected,taking evaluations on p rational places which are splitting completely;finally,we construct binary sequences with length of odd primes p by using quadratic character,transform the auto-and cross-correlations of binary sequences into the estimation of the number of rational points on Kummer curve,and obtain the upper bound of correlation from Serre upper bound.