PSO-Guided Construction of MRD Codes for Rank Metrics

Behnam Dehghani () and Amineh Sakhaie
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Behnam Dehghani: Department of Mechanical Engineering, University of Minho, Campus of Azurém, 4800-058 Guimarães, Portugal
Amineh Sakhaie: Mathematics Department, Faculty of Science, Lisbon University, 1649-004 Lisbon, Portugal

Mathematics, 2025, vol. 13, issue 17, 1-13

Abstract: 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.

Keywords: maximum rank-distance codes; Particle Swarm Optimization; rank metrics; error-correcting codes; combinatorial optimization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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