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

2025-11-24

AI Summary

This paper addresses the controllable construction of linear codes with prescribed hull dimension—i.e., the intersection of a code and its dual—over finite fields. To overcome the difficulty of precisely controlling the Euclidean hull dimension, the authors propose two novel methods: (i) they prove for the first time that any LCD code (hull dimension zero) with minimum distance at least 2 can be transformed via monomial equivalence into a code with one-dimensional hull; (ii) they establish a dimension-lifting mechanism to construct an $(ell+1)$-dimensional hull code from an $ell$-dimensional one, breaking the traditional reliance on dimension-reduction techniques. Leveraging duality analysis, projection mappings, and equivalence transformations under mild conditions, the approach enables explicit, fine-grained control over hull dimension. Several constructive families of $ell$-dimensional hull codes are provided, supporting reverse conversion from higher- to lower-dimensional hulls. This significantly expands the systematic construction framework for small-hull codes and enhances their applicability in coding theory and cryptography.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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.

Problem

• Investigates hull construction for linear codes over finite extensions of binary fields
• Explores equivalence between LCD codes and one-dimensional hull codes
• Develops methods to construct higher-dimensional hulls from lower-dimensional ones

Innovation

• Constructs one-dimensional hull codes over finite fields
• Builds l+1-dimensional hulls from l-dimensional hulls
• Derives multiple constructions for linear code hulls