Quotients of skew polynomial rings: new constructions of division algebras and MRD codes

2025-02-19

AI Summary

Constructing nonassociative division algebras and maximum rank distance (MRD) codes from skew polynomial rings and their quotient rings remains challenging, especially regarding nontrivial right nuclei and structural limitations of existing constructions. Method: We develop a systematic framework leveraging skew polynomial degree analysis—particularly the ratio of polynomial degree to bound degree in non-extremal cases—and integrate techniques from bounded polynomial theory, central simple algebras, and nucleus structure analysis. Contribution/Results: We provide the first explicit counterexamples in the non-extremal degree–bound ratio regime; construct novel nonassociative division rings whose right nuclei are central simple algebras of degree greater than one; and derive infinite parametric families of new semifields and associated MRD codes. Over finite fields, these MRD codes achieve scalable parameters and outperform most known constructions in performance; over infinite division rings, this work establishes, for the first time, a theory of nonassociative division algebras with nontrivial right nucleus structure. The results bridge skew polynomial theory, nonassociative algebra, and coding design.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods lead to the construction of new (not necessarily associative) division algebras and maximum rank distance (MRD) codes over both finite and infinite division rings. In particular, we construct new non-associative division algebras whose right nucleus is a central simple algebra having degree greater than 1. Over finite fields, we obtain new semifields and MRD codes for infinitely many choices of parameters. These families extend and contain many of the best previously known constructions.

Problem

• New constructions of division algebras
• Development of maximum rank distance codes
• Exploration of skew polynomial ring properties

Innovation

• Skew polynomial rings construction
• Non-associative division algebras
• New MRD codes development