Linear codes constructed from defining sets have been extensively studied since they may have good parameters if the defining sets are chosen properly. Let \(\mathbb{F}_{p^m}\) be the finite field with \(p^m\) elements, where p is an odd prime and m is a positive integer. In this paper, we study the linear code \({\mathcal {C}}_D=\{ (\mathrm{Tr}(\alpha x))_{x \in D}\, |\, \alpha \in {\mathbb {F}}_{p^m}\}\) by choosing the defining set \(D=\{x \in {\mathbb {F}}_{p^m}^*\, | \, \mathrm{Tr}(ax^2+bx)=0\}\), where \(a\in {\mathbb {F}}_{p^m}^*\) and \(b \in {\mathbb {F}}_{p^m}\). Several classes of linear codes with explicit weight distribution are obtained. The parameters of some proposed codes are new. Several examples show that some of our codes are optimal or almost optimal according to the tables of best codes known in Grassl. Our results generalize some results in Ding and Ding (IEEE Trans. Inf. Theory 61(11):5835–5842, 2015), Li et al. (Disc. Math. 241:25–38, 2018).

由定义集构造的线性代码已被广泛研究，因为如果正确选择定义集，它们可能具有良好的参数。设\(\mathbb{F}_{p^m}\)是具有\(p^m\) 个元素的有限域，其中p是奇素数，m是正整数。在本文中，我们研究线性代码\({\mathcal {C}}_D=\{ (\mathrm{Tr}(\alpha x))_{x \in D}\, |\, \alpha \in {\mathbb {F}}_{p^m}\}\)通过选择定义集\(D=\{x \in {\mathbb {F}}_{p^m}^*\, | \ , \mathrm{Tr}(ax^2+bx)=0\}\)，其中\(a\in {\mathbb {F}}_{p^m}^*\)和\(b \in { \mathbb {F}}_{p^m}\). 获得了几类具有显式权重分布的线性码。一些建议代码的参数是新的。几个例子表明，根据 Grassl 中已知的最佳代码表，我们的一些代码是最佳的或几乎是最佳的。我们的结果概括了 Ding 和 Ding (IEEE Trans. Inf. Theory 61(11):5835–5842, 2015)、Li 等人的一些结果。（Disc. Math. 241:25–38, 2018）。