Shortest Embeddings of Linear Codes with Arbitrary Hull Dimension
This work addresses a theoretical coding theory problem for researchers, providing incremental extensions to known embedding methods.
The paper tackles the problem of finding shortest embeddings of linear codes with arbitrary hull dimensions in Euclidean and Hermitian cases, extending previous work on LCD and self-orthogonal embeddings. It obtains exact lengths for these embeddings using quadratic form theory and group theory, improves prior results on self-orthogonal embeddings, and produces optimal codes not in existing databases.
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.