The Weight Distributions Of A Class Of Cyclic Codes

In order to correct the mistakes in the process of information transmission, people need to encode information. By using error correcting code to inspect or correct the received information, one can improve the quality of information. Cyclic codes are a kind of important linear codes, and it has rich algebraic structure and many of the simple and practical algorithm of decoding, At the same time,their algorithms of encoding and decoding can be easily implemented by linear feedback shift register. The weight distribution of the cyclic code refers to the number of distinct weights of the code and the corresponding multiplicity of each weight, the weight distribution of a code not only gives the error correcting ability of the code, but also allows the computation of the error probability of error detection and correction. Therefore, the weight distribution of cyclic code is always an important hot topic in the field of coding theory.This paper first introduces a class of non-primitive cyclic code with two nonzeros, and sums up the known results on weight distribution of this kind of non-primitive cyclic code, then we obtain the weight distribution of a new class of non-primitive cyclic code two with nonzeros. By some known results on Gaussian sums and Davenport-Hasse Theorem we obtain the explicit formula on the square of quartic Gaussian sums over finite fields, and then all desired Jacobi sums are derived. Finally, the weight distribution of the code is obtained by evaluating some exponential sums.