Designs, Codes and Cryptography

, Volume 83, Issue 1, pp 83–99 | Cite as

Complete weight enumerators of a class of linear codes



Let \(\mathbb {F}_{q}\) be the finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. For a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and let \({\mathrm {Tr}}_{m}\) be the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}\setminus \{(0,0,\ldots ,0)\} : {\mathrm {Tr}}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\},\) we define a p-ary linear code \(\mathcal {C}_{D}\) by
$$\begin{aligned} \mathcal {C}_{D}=\{\mathbf {c}(a_{1},a_{2},\ldots ,a_{t}) : (a_{1},a_{2},\ldots ,a_{t})\in \mathbb {F}^{t}_{q}\}, \end{aligned}$$
$$\begin{aligned} \mathbf {c}(a_{1},a_{2},\ldots ,a_{t})=({\mathrm {Tr}}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2}+\cdots +a_{t}x^{2}_{t}))_{(x_{1},x_{2},\ldots ,x_{t}) \in D}. \end{aligned}$$
We shall present the complete weight enumerators of the linear codes \(\mathcal {C}_{D}\) and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016).


Linear codes Weight distribution Gauss sums 

Mathematics Subject Classification

94B05 11T23 11T71 



The authors would like to express deepest thanks to the editor and the anonymous reviewers for their invaluable comments and suggestions to improve the quality of this paper. Without their careful reading and sophisticated advice, the paper would have never been developed like this.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsChungnam National UniversityDaejeonKorea
  2. 2.School of Computer Science and Software EngineeringEast China Normal UniversityShanghaiChina
  3. 3.Department of MathematicsKAISTDaejeonKorea

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