Designs, Codes and Cryptography

, Volume 83, Issue 1, pp 83–99

# Complete weight enumerators of a class of linear codes

• Jaehyun Ahn
• Dongseok Ka
• Chengju Li
Article

## Abstract

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}
where
\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).

## Keywords

Linear codes Weight distribution Gauss sums

## Mathematics Subject Classification

94B05 11T23 11T71

## Notes

### Acknowledgments

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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