The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.

有限域上的线性代码的外壳是代码与其对偶的交集，这是由 Assmus 和 Key 引入的。在本文中，我们开发了一种利用高斯和的正特征类比构造具有平凡包（LCD 码）和一维包的线性代码的方法。这些代码是准阿贝尔的，有时是双重循环的。给出了线性代码成为 LCD 代码（或具有一维外壳的线性代码）的一些充分条件。值得一提的是，我们提出了构建的线性代码的最小距离的下限。作为应用程序，使用这些条件，我们获得了关于 Grassl 在线数据库的一些最佳或几乎最佳的 LCD 代码（分别是具有一维外壳的线性代码）。