PSO-Guided Construction of MRD Codes for Rank Metrics

• Published in: Mathematics (Basel) Vol. 13; no. 17; p. 2756
• Main Authors: Dehghani, Behnam, Sakhaie, Amineh
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2025
• Subjects: Algebra; Algorithms; Analysis; Codes; combinatorial optimization; Convolutional codes; Error correcting codes; Error correction; Error correction & detection; Fields (mathematics); Genetic algorithms; Heuristic; Heuristic methods; Machine learning; Mathematical optimization; maximum rank-distance codes; Optimization; Parameters; Particle Swarm Optimization; Python; rank metrics; Source code; Germany
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math13172756

Maximum Rank-Distance (MRD) codes are a class of optimal error-correcting codes that achieve the Singleton-like bound for rank metric, making them invaluable in applications such as network coding, cryptography, and distributed storage. While algebraic constructions of MRD codes (e.g., Gabidulin codes) are well-studied for specific parameters, a comprehensive theory for their existence and structure over arbitrary finite fields remains an open challenge. Recent advances have expanded MRD research to include twisted, scattered, convolutional, and machine-learning-aided approaches, yet many parameter regimes remain unexplored. This paper introduces a computational optimization framework for constructing MRD codes using Particle Swarm Optimization (PSO), a bio-inspired metaheuristic algorithm adept at navigating high-dimensional, non-linear, and discrete search spaces. Unlike traditional algebraic methods, our approach does not rely on prescribed algebraic structures; instead, it systematically explores the space of possible generator matrices to identify MRD configurations, particularly in cases where theoretical constructions are unknown. Key contributions include: (1) a tailored finite-field PSO formulation that encodes rank-metric constraints into the optimization process, with explicit parameter control to address convergence speed and global optimality; (2) a theoretical analysis of the adaptability of PSO to MRD construction in complex search landscapes, supported by experiments demonstrating its ability to find codes beyond classical families; and (3) an open-source Python toolkit for MRD code discovery, enabling full reproducibility and extension to other rank-metric scenarios. The proposed method complements established theory while opening new avenues for hybrid metaheuristic–algebraic and machine learning–aided MRD code construction.