On Construction of Linear (Euclidean) Hull Codes over Finite Extensions Binary Fields

TL;DR

This paper investigates the construction and properties of linear hulls of codes over finite fields, especially focusing on one-dimensional hulls and methods to generate higher-dimensional hulls from lower-dimensional ones.

Contribution

It introduces new methods for constructing hulls of various dimensions from existing codes, particularly for LCD codes over extended binary fields, under weak conditions.

Findings

Any LCD code over FF_q (q > 3) with minimum distance  2 is equivalent to a code with a one-dimensional hull.

Provides a construction method to generate (\u0013 + 1)-dimensional hulls from -dimensional hulls.

Derives multiple new constructions for -dimensional hulls of linear codes.

Abstract

The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to check the permutation equivalence of two linear codes and compute a linear code's automorphism group. Research has shown that these algorithms are very effective when the hull size is small. Linear complementary dual (LCD) codes have the smallest hulls, while codes with a one-dimensional hull have the second smallest. A recent notable paper that directs our investigation is authored by H. Chen, titled ``On the Hull-Variation Problem of Equivalent Linear Codes", published in IEEE Transactions on Information Theory, volume 69, issue 5, in 2023. In this paper, we first explore the one-dimensional hull of a linear code over finite fields. Additionally, we demonstrate that any LCD code over an extended binary field \( \FF_q \) (where \( q > 3 \)) with a minimum distance of at least $2$ is equivalent to the one-dimensional hull of a linear code under a specific weak condition. Furthermore, we provide a construction for creating hulls with \( \ell + 1 \)-dimensionality from an \( \ell \)-dimensional hull of a linear code, again under a weak condition. This corresponds to a particularly challenging direction, as creating \( \ell \)-dimensional hulls from \( \ell + 1 \)-dimensional hulls. Finally, we derive several constructions for the \( \ell \)-dimensional hulls of linear codes as a consequence of our results.