A Construction of Linear Codes and Their Complete Weight Enumerators

Recently, linear codes constructed from defining sets have been studied extensively. They may have nice parameters if the defining set is chosen properly. Let $m >2$ be a positive integer. For an odd prime $p$, let $r=p^m$ and $\text{Tr}$ be the absolute trace function from $\mathbb{F}_r$ onto $\mathbb{F}_p$. In this paper, we give a construction of linear codes by defining the code $C_{D}=\{(\mathrm{Tr}(ax))_{x\in D}: a \in \mathbb{F}_{r} \},$ where $D =\left\{x\in \mathbb{F}_{r} : \mathrm{Tr}(x)=1, \mathrm{Tr}(x^2)=0 \right\}.$ Its complete weight enumerator and weight enumerator are determined explicitly by employing cyclotomic numbers and Gauss sums. In addition, we obtain several optimal linear codes with a few weights. They have higher rate compared with other codes, which enables them to have essential applications in areas such as association schemes and secret sharing schemes.