Complete weight enumerators of a class of linear codes

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).