Propagation rules are of great help in constructing good linear codes. Both Euclidean and Hermitian hulls of linear codes perform an important part in coding theory. In this paper, we consider these two aspects together and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for resulting codes derived from these two propagation rules being self-dual, self-orthogonal or linear complement dual (LCD) codes are given. As applications, we construct some linear codes with prescribed hull dimensions and many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD codes and good quaternary Hermitian LCD codes which are optimal or have best or almost best known parameters according to Datebase at $http://www.codetables.de$. Moreover, our methods contributes positively to improve the lower bounds on the minimum distance of known LCD codes.

翻译：推广规则对构建良好的线性代码大有帮助。 Euclidean 和 Hermitian 的线性代码壳在编码理论中扮演重要角色。 在本文中,我们共同考虑这两个方面,并确定Euclidean 和 Hermitian 的两个古典传播规则,即直接总和构造和$(mathbf{u},\mathbf{u+v}) 美元构建。由这两个传播规则产生的新代码标准是自成一体的、自成一体或线性补充双重(LCD)代码。作为应用,我们制定了一些带有规定的船体尺寸的线性代码,以及许多新的二进制、永恒的Euclidean 正式的自成二进制(FSD)代码、四进制的Hermitian FSD LCD 代码以及良好的Kemtinary Hermitian LCD 代码,这些代码是最佳的,或者根据日期数据库在 $/www.codebtables.de$。 此外,我们的方法有助于改善已知最低距离的LCD代码。