The Weight Distribution Of Cyclic Codes With Two Non-zeros

Cyclic codes are one of the most important linear codes.They have good algebraic structure and properties,as well as very efficient encoding and decoding algorithms.Therefore,cyclic codes are widely used in communication system.Since the study on the weight distribution of a cyclic code can improve decoding efficiency and the error correction ability,the research on the weight distribution of cyclic codes is an important topic in coding theory.In this thesis,we mainly study the weight distribution of three classes of cyclic codes with two non-zeros by the knowledge of quadratic form theory and exponential sum on finite fields and some other related knowledge.Let p be an odd prime,m be odd,a be the primitive element of the finite field Fpm.The cyclic codes in this paper are as follows:C = {(c0,c1…,cL-1)| ci = Tr1m(axir + bxi),0?i?L-1},where L = pm-1,a,b ? Fpm,xi?Fpm*.When r is equal to l,t,d respectively and satisfies different conditions,the following main results are obtained in this paper:1.when p ? 3(mod 4),we determine the weight distribution of cyclic codes with non zeros ?-1,?-1 over Fp and prove they have three weights;2.when p = 3(mod 4),we obtain the weight distribution of cyclic codes with non zero ?-1,?-t over Fp and prove they have three weights;3.when p ? 1(mod 4),we get the weight distribution of cyclic codes with non-zero points ?-1,?-d over Fp and prove they have three weights.These results above generalize the existing results of the relevant literature.