Research On Several Classes Of Cyclic Codes Over Finite Field

As a subclass of linear codes,cyclic codes not only have a good algebraic structure,but also have efficient coding and decoding algorithms.Therefore,it is widely used in research fields such as secret sharing schemes,authentication codes,combining schemes,design of frequency hopping sequences and distributed storage.Therefore,it is of great significance to study and construct cyclic codes with good properties.The main results of this article are as follows:Firstly,this paper discusses a class of binary cyclic codes with two nonzeros.By using the Gauss period over finite field,the number and exponential sum of the solution of the equation,the parameters of such binary cyclic codes are calculated as[2^m-1,2m,2^m-1-2^k],and its weight distribution,it is a class of binary four-weight cyclic codes.At the same time,such binary cyclic codes can be used to construct secret sharing schemes.Secondly,this paper also constructs two classes of ternary cyclic codes.By using the quadratic character over finite fields,the factorization of polynomials and the structure of the equation solutions,the parameters of the two classes of ternary cyclic codes are[3^m-1,3^m-2m-1,4],and they are optimal with respect to the Sphere-Packing theory.