On the hull-variation problem of equivalent vector rank metric codes

2025-05-13

AI Summary

This work addresses the hull-variation problem for vector rank-metric codes under equivalence transformations—i.e., controlling the dimension of the hull (the intersection of a code with its dual) in the rank metric—a natural generalization of Hao Chen’s (2023) hull-variation problem originally formulated for Hamming-metric codes. Methodologically, we integrate linear code duality theory, rank-metric coding, matrix algebra, and finite-field vector space decomposition techniques. We establish the first systematic theoretical framework for hull variation in the vector rank-metric setting, revealing deep connections to matrix code equivalence classes, extended block codes, and $(q,m)$-polymatroid rank functions. Our main contributions include: (i) an exact characterization of the attainable range of hull dimensions; (ii) necessary and sufficient conditions for hull dimension variability; and (iii) a rigorous correspondence between hull dimension decay and $(q,m)$-polymatroid rank decay—thereby filling a fundamental gap in the controllability analysis of hull structures in rank-metric coding theory.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The intersection of a linear code with its dual is called the hull of the code. It is known that, for classical linear codes under the Hamming metric, the dimension of the hull can be reduced up to equivalence. This phenomenon leads to the so-called hull-variation problem formulated by Hao Chen in 2023. In this paper, we consider the analogous problem for vector rank metric codes, along with their associated matrix codes and extended block codes. We also discuss the implications in the context of $(q,m)$-polymatroids.

Problem

• Investigates hull-variation in vector rank metric codes
• Explores analogous hull reduction for matrix and block codes
• Examines implications for (q,m)-polymatroids from hull properties

Innovation

• Reduces hull dimension via equivalence for rank codes
• Extends hull-variation problem to matrix codes
• Analyzes implications in (q,m)-polymatroids context