Shortest Embeddings of Linear Codes with Arbitrary Hull Dimension
2026-04-10 • Information Theory
Information Theory
AI summaryⓘ
The authors study how to find the shortest way to represent linear codes with a certain property called the "t-dimensional hull" in two mathematical settings: Euclidean and Hermitian. They extend previous work that focused only on special cases to cover any hull dimension and any finite field. Using theories from quadratic forms and group theory, they classify linear codes into types and give methods to find these shortest representations. They improve earlier results and give new examples, including some optimal codes not found in existing databases.
linear codeshull dimensionEuclidean embeddingHermitian embeddingquadratic formsfinite fieldsGram matrixself-orthogonal codeLCD codesclassical groups
Authors
Jiabin Wang, Jinquan Luo
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.