For any positive integer m > 2 and an odd prime p, let \(\mathbb {F}_{p^{m}}\) be the finite field with p^m elements and let \( \text {Tr}^{m}_{e}\) be the trace function from \(\mathbb {F}_{p^{m}}\) onto \(\mathbb {F}_{p^{e}}\) for a divisor e of m. In this paper, for the defining set \(D=\{x\in \mathbb {F}_{p^{m}}:\text {Tr}^{m}_{e}(x)=1\text { and } \text {Tr}^{m}_{e}(x^{2})=0\}=\{d_{1}, d_{2}, \ldots , d_{n}\}\)(say), we define a p^e-ary linear code \(\mathcal {C}_{D}\) by

Then we determine the complete weight enumerator and weight distribution of the linear code \(\mathcal {C}_{D}\). The presented code is optimal with respect to the Griesmer bound provided that \(\frac {m}{e}=3\). In fact, it is MDS when \(\frac {m}{e}=3\). This paper gives the results of S. Yang, X. Kong and C. Tang (Finite Fields Appl. 48 (2017)) if we take e = 1. In addition to the generalization of the results of Yang et al., we study the dual code \(\mathcal {C}_{D}^{\perp }\) of the code \(\mathcal {C}_{D}\) as well as find some optimal constant composition codes.

对于任何正整数m > 2 和奇素数p，令\(\mathbb {F}_{p^{m}}\)是具有p ^{m 个}元素的有限域，并令\( \text {Tr}^{米} _ {E} \）是从跟踪函数\（\ mathbb {F} _ {p ^ {米}} \）到\（\ mathbb {F} _ {p ^ {E}} \）用于除数Ë的米。在本文中，对于定义集\(D=\{x\in \mathbb {F}_{p^{m}}:\text {Tr}^{m}_{e}(x)=1\文本 { 和 } \text {Tr}^{m}_{e}(x^{2})=0\}=\{d_{1}, d_{2}, \ldots , d_{n}\} \） （比方说），我们定义一个p ^ë进制线性码\（\ mathcal {C} _ {d} \）由

然后我们确定线性代码\(\mathcal {C}_{D}\)的完整权重枚举器和权重分布。如果\(\frac {m}{e}=3\) ，所提供的代码对于 Griesmer 边界是最佳的。实际上，当\(\frac {m}{e}=3\)时就是 MDS 。本文给出了 S. Yang、X. Kong 和 C. Tang (Finite Fields Appl. 48 (2017)) 的结果，如果我们取e = 1。除了对 Yang 等人的结果进行推广，我们研究双码\（\ mathcal {C} _ {d} ^ {\ PERP} \）的代码\（\ mathcal {C} _ {d} \）以及找到一些最佳恒定的组成码。