Shortest Embeddings of Linear Codes with Arbitrary Hull Dimension

TL;DR

This paper determines the shortest embeddings of linear codes with arbitrary hull dimensions over finite fields, extending prior work on LCD and self-orthogonal codes using quadratic form theory.

Contribution

It introduces a comprehensive classification and constructive algorithms for shortest hull embeddings of linear codes with arbitrary hull dimensions over finite fields.

Findings

Exact lengths of shortest hull embeddings obtained.

Classification of codes into types based on Gram matrix congruence.

Constructed examples of optimal codes not in existing databases.

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.