A Construction of Linear Codes and Their Complete Weight Enumerators

Abstract: 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.