The Weight Distributions Of Several Class Of Cyclic Codes

Cyclic codes are a special class of linear codes with very preferable algebraic properties, they have been widely used in data storage, the structure of quantum code, the structure of frequency hopping sequence and so on. The weight distribution of a code not only gives the error correcting ability of the code, but also can calculate the error incidence rate of information transmission, cyclic codes have been widely applied to inspect or correct the received information. It has become an important way to improve the quality of information transmission. Therefore, researching the weight distribution of cyclic codes have theoretical significance and application value.The weight distribution of cyclic codes always an important aspect of cyclic codes research. In recent years, cyclic codes are mostly studied in coding theory. In this pa-per, we structure some cyclic codes with three or four nonzeros over finite field. Using the theory of quadratic forms, especially, the connection between the type and rank of quadratic forms, MacWilliamse identities and some special algebratic equations’solv-ing skills, we determined the weight distribution of these code, calculates the weight distribution of some cyclic codes with three or four nonzeros. When p is odd prime,we have calculated the weight distribution of p-ary cyclic codes, which has nonze-ro·α-2,α-(p2k+1),α-(p4k+1);·α-1,α-(pk+1),α-(p3k+1);·±α-1,α-t;α-1,±α-t,±α-1,±α-t And we calculated the weight distribution of two classes of binary cyclic codes, which has nonzero·α-1,α-(2k+1),α-(2k+1);·α-1,α-(2k+1),α-(23k+1) where a is a primitive element of finite field, k and t are some positive with certain conditions.