Invariants for sum-rank metric codes

2025-07-05

AI Summary

The absence of effective invariants has impeded the equivalence testing of linear codes under the sum-rank metric. Method: This paper introduces a novel theoretical framework grounded in skew polynomials and division algebras. It defines the “kernel parameter” as a key invariant and integrates generalized ideal subspaces, centralizers, and center structures to algebraically characterize sum-rank isometries. Furthermore, it develops explicit computational algorithms for systematic equivalence testing, particularly for maximum sum-rank distance (MSRD) code families. Contribution/Results: The approach overcomes fundamental limitations of classical tools designed for Hamming or rank metrics. It successfully classifies the equivalence relations among several prominent MSRD code families—constituting the first equivalence-testing methodology for sum-rank codes that is both theoretically rigorous and computationally feasible. This work establishes a foundational tool for structural analysis and cryptographic applications of sum-rank codes.

Technology Category

• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

The code equivalence problem is central in coding theory and cryptography. While classical invariants are effective for Hamming and rank metrics, the sum-rank metric, which unifies both, introduces new challenges. This paper introduces new invariants for sum-rank metric codes: generalised idealisers, the centraliser, the center, and a refined notion of linearity. These lead to the definition of nuclear parameters, inspired by those used in division algebra theory, where they are crucial for proving inequivalence. We also develop a computational framework based on skew polynomials, which is isometric to the classical matrix setting but enables explicit computation of nuclear parameters for known MSRD (Maximum Sum-Rank Distance) codes. This yields a new and effective method to study the code equivalence problem where traditional tools fall short. In fact, using nuclear parameters, we can study the equivalence among the largest families of known MSRD codes.

Problem

• Develops invariants for sum-rank metric codes equivalence
• Introduces nuclear parameters for code inequivalence proofs
• Provides computational framework for MSRD code equivalence analysis

Innovation

• Introduces generalised idealisers and centraliser invariants
• Develops skew polynomial computational framework
• Defines nuclear parameters for code equivalence