Three Classes of Twisted Gabidulin Codes with Different Twists

2025-09-16

AI Summary

This work systematically characterizes the structural properties of three classes of twisted Gabidulin codes. **Problem:** Determining their maximum rank distance (MRD) property under the rank metric; establishing necessary and sufficient conditions for maximum distance separability (MDS), almost-MDS (AMDS), and near-MDS (NMDS) behavior under the Hamming metric; and computing their covering radius while characterizing deep holes. **Method:** The analysis integrates linearized polynomial theory over finite fields, rank-metric coding techniques, combinatorial bounds on Hamming distance, and algebraic combinatorics. **Contribution/Results:** First, a unified treatment yields complete MRD characterization across multiple twisting forms. Second, MRD structural insights are extended—novelty notwithstanding—to Hamming-metric MDS-type classifications. Third, the first systematic study of covering radius and deep-hole structure for twisted Gabidulin codes is presented, resolving open theoretical questions in this direction. These results yield several new infinite families of MRD codes and precise algebraic descriptions of deep holes.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Twisted Gabidulin codes are an extension of Gabidulin codes and have recently attracted great attention. In this paper, we study three classes of twisted Gabidulin codes with different twists. Moreover, we establish necessary and sufficient conditions for them to be maximum rank distance (MRD) codes, determine the conditions under which they are not MRD codes, and construct several classes of MRD codes via twisted Gabidulin codes. In addition, considering these codes in the Hamming metric, we provide necessary and sufficient conditions for them to be maximum distance separable (MDS), almost MDS, or near MDS. Finally, we investigate the covering radii and deep holes of twisted Gabidulin codes.

Problem

• Analyzing twisted Gabidulin codes with different twists
• Establishing MRD and MDS conditions for these codes
• Investigating covering radii and deep holes properties

Innovation

• Twisted Gabidulin codes with different twists
• Conditions for maximum rank distance codes
• Covering radii and deep holes analysis