Weight enumerators of a class of linear codes

Abstract

Recently, linear codes constructed from defining sets have been studied widely and they have many applications. For an odd prime p, let $q=p^{m}$ for a positive integer m and $\mathrm {Tr}_{m}$ the trace function from $\mathbb {F}_{q}$ onto $\mathbb {F}_{p}$. In this paper, for a positive integer t, let $D\subset \mathbb {F}^{t}_{q}$ and $D=\{(x_{1},x_{2}) \in (\mathbb {F}_{q}^{*})^{2} : \mathrm {Tr}_{m}(x_{1}+x_{2})=0\}$, we define a p-ary linear code $\mathcal {C}_{D}$ by

$$\begin{aligned} \mathcal {C}_{D}=\left\{ \mathbf {c}(a_{1},a_{2}) : (a_{1},a_{2})\in \mathbb {F}^{2}_{q}\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathbf {c}(a_{1},a_{2})=\left( \mathrm {Tr}_{m}\left( a_{1}x^{2}_{1}+a_{2}x^{2}_{2}\right) \right) _{(x_{1},x_{2})\in D}. \end{aligned}$$

We compute the weight enumerators of the punctured codes $\mathcal {C}_{D}$.