In this paper, we study the shortest $t$-dimensional hull embeddings of linear codes in both Euclidean and Hermitian cases, extending the existing research on the shortest LCD and self-orthogonal embeddings to arbitrary hull dimensions and arbitrary finite fields. We obtain the exact length of such embeddings by adopting tools from quadratic form theory over finite fields and classical group theory. Based on the congruence equivalence class of Gram matrices of linear codes, we classify linear codes into distinct ``types'' and present corresponding constructive algorithms. In particular, we improve the results of An et al. and fully determine the length of the shortest self-orthogonal embeddings for linear codes. Finally, applying these algorithms, we provide examples for various settings and obtain several optimal codes inequivalent to those in the BKLC database.

翻译：本文研究欧几里得情形与埃尔米特情形下线性码的$t$维壳最短嵌入，将现有关于最短LCD码与自正交嵌入的研究推广至任意壳维数与任意有限域。通过采用有限域上二次型理论与经典群论的工具，我们获得了此类嵌入的精确长度。基于线性码Gram矩阵的合同等价类，我们将线性码划分为不同“类型”，并给出了相应的构造性算法。特别地，我们改进了An等人的结果，完整确定了线性码最短自正交嵌入的长度。最后，应用这些算法，我们提供了多种参数下的实例，并得到了若干与BKLC数据库中不等价的最优码。