Construction Of Several Classes Of Linear Codes With A Few Weights And Their Parameter Analysis

Linear codes have important applications in many aspects,so the construction of linear codes has attracted the attention of many scholars,and has always been a very important research target in error-correcting coding theory.Especially,it is very important to the research on linear codes with a few weights for the research of linear codes.Linear codes that have good performance with a few weights have good applications in many fields.For example,they have very important applications in combinatorial design and secret sharing schemes.The weight distribution of linear codes contains a lot of basic information about codes.However,for general linear codes,it is very difficult to determine the minimum weight of the linear codes.Only a few very special linear codes can determine their weight distribution.One of the most important methods to design linear codes with a few weights is based on the construction of the defining set.If the defining set is properly selected,some optimal codes and almost optimal codes with a few weights can be obtained.In this paper,based on the theory of the trace function in finite fields and the theory of defining set,by selecting defining set properly,several kinds of linear codes with a few weights over the odd prime finite fields are constructed.Then,we determine the length,dimension and weight of the codes by using the orthogonal property of the additive character,multiplicative character and the Guass sums over the finite fields.We also design Magma programs to verify the correctness of our result.The main work is as follows:(1)We design and obtain a class of linear codes with a few weights for the defin-ing set be the preimage of a general function under two trace function mappings,which are four-weight and six-weight respectively.The parameters and weight enu-merator of the linear codes are studied and determined.From the results,the codes we constructed and obtained in this chapter include some optimal codes with re-spect to the codetables and some almost optimal codes with respect to the Griesmer Bound.(2)We construct several classes of linear codes with a few weights for the defining set be the intersection of the preimage of a general function and quadratic residue under the trace function,which are three weights,four weights and six weights respectively.And their parameters and weight enumerator of the linear codes that we obtained are also studied and determined.The results show that the codes we obtained include some optimal codes with respect to the Singleton Bound,the Griesmer Bound and the codetables in this chapter.It is worth noting that the linear codes constructed in this chapter can be used to construct the puncturing codes.