🤖 AI Summary
This study addresses the problem of determining the shortest embeddings of linear codes with arbitrary hull dimensions under both Euclidean and Hermitian inner products. By leveraging the theory of quadratic forms over finite fields, classical group theory, and the congruence classification of Gram matrices, the authors develop a unified framework that generalizes prior results on shortest LCD codes and self-orthogonal embeddings to arbitrary hull dimensions and arbitrary finite fields. Within this framework, they completely characterize the minimal length of self-orthogonal embeddings for any given linear code, thereby improving upon existing bounds in the literature. Furthermore, through an explicit constructive algorithm, they obtain multiple new optimal linear codes that surpass the best-known entries in the BKLC database.
📝 Abstract
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.