The Research Of Two Classes Of Binary Cyclic Codes And Their Duals

As a subclass of linear codes,cyclic codes have important applications in consumer electronics,data storage systems,authentication systems,secret sharing schemes and communication systems for efficient encoding and decoding algorithms compared with the linear block codes.The weight distribution of cyclic codes determines the minimal distance and the error correcting capability of codes.Thus,It is of great significance to study the weight distribution of cyclic codes.During the past few decades,cyclic codes have attracted a lot of attention and much progress has been made.A main objective of this thesis is to present two classes of cyclic codes with two zeros.It turns out that these two classes of cyclic codes are optimal or almost optimal with respect to the Sphere Packing bound.The main contents of this thesis are as follows:In Chapter one and Chapter two,we mainly introduce the background and known works on cyclic codes,and some preliminaries needed later.In Chapter three,we introduce some results about Kloosterman sums and study the relationship between some specific Kloosterman sums and Gaussian periods.By solving some equations in GF(2t),some distributions of Kloosterman sums are completely determined.In Chapter four,we present two classes of cylcic codes and determine the weight distributions of their duals.Let m=2tk and α be a primitive element of the finite field GF(2m)for positive t and k>2.For t=4,the first class of cyclic codes has generator polynomial g1(x)g2m-1/17(x)where gi(x)denotes the minimal polynomial of αi over GF(2).It turns out that this class of cyclic codes has parameters[2m-1,2m-m-9,4],which is distance-optimal with respect to Sphere Packing bound.When t=5,the second class of cyclic codes has generator polynomial g1(x)g 2m-1/33(x).The weight distribution of the duals of these two classes of cyclic codes are completely determined based on some results on Gaussian periods and Kloosterman sums.