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.

翻译：Assmus 和 Key 引入了该代码及其双元的交汇点。 在本文中,我们开发了一种方法,通过使用高斯数值的正性特征类比,用浅质体(LCD代码)和一维体来构建线性代码。这些代码是准酸的,有时是双环状的。提出了线性代码作为LCD代码(重写一维体的线性代码)的足够条件。值得一提的是,我们在构建的线性代码的最低距离上设定了较低的约束。作为应用,我们利用这些条件获得了某些最佳或几乎最佳的LCD代码(用一维体制的线性代码重印线性代码 ) 。