PSO-Guided Construction of MRD Codes for Rank Metrics

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. 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 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 PSO formulation that encodes rank-metric constraints into the optimization process, ensuring feasible code candidates; (2) empirical evidence supporting the existence of MRD codes beyond classical families, with analysis of their properties; and (3) an open-source Python toolkit for MRD code discovery, enabling reproducibility and extension to other code parameters. Our experiments demonstrate the efficacy of PSO in identifying MRD codes with novel parameters, bridging gaps in the algebraic literature. The proposed method not only complements theoretical approaches but also opens avenues for machine learning-aided code design in future research.