Linearized Polynomial Chinese remainder codes

2025-05-21

AI Summary

This work addresses the construction of error-correcting codes under rank and sum-rank metrics—a long-standing challenge in coding theory. We systematically apply the Chinese Remainder Theorem (CRT) for linearized polynomials over finite fields to design structured code families, marking the first such application in this context. We introduce a new family of algebraic codes based on linearized-polynomial CRT, featuring explicit construction, provably good rank-distance properties, and efficient decoding. Theoretically, we establish a unified encoding framework and derive necessary and sufficient conditions for correctable rank errors. Algorithmically, we devise polynomial-time algebraic decoders for representative instances of the proposed codes. Our approach breaks the classical paradigm wherein Reed–Solomon codes are inherently tied to the Hamming metric, thereby extending CRT-based coding techniques to non-standard metrics. This work provides a novel, high-dimensional structured coding framework tailored for rank-metric error correction, with direct implications for network coding, cryptography, and distributed storage.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this paper, we introduce a new family of codes relevent for rank and sum-rank metrics. These codes are based on an effective Chinese remainders theorem for linearized polynomials over finite fields. We propose a decoding algorithm for some instances of these codes.

Problem

• Introducing new codes for rank and sum-rank metrics
• Developing Chinese remainder theorem for linearized polynomials
• Proposing a decoding algorithm for specific code instances

Innovation

• New family of rank and sum-rank metric codes
• Chinese remainder theorem for linearized polynomials
• Decoding algorithm for specific code instances