Skew polynomial representations of matrix algebras and applications to coding theory

2025-12-09

AI Summary

Existing constructions of Maximum Sum-Rank Distance (MSRD) codes suffer from structural limitations and lack a unifying algebraic framework applicable across finite fields, infinite fields, and division rings. Method: We introduce a unified algebraic framework based on skew polynomial representations, extending matrix algebra’s skew polynomial representation to direct sums of matrix spaces over division rings. We establish, for the first time, an exact correspondence between rank distance and both the degree of skew polynomials and their greatest common right divisor. Our approach integrates skew polynomial ring theory, algebraic coding, matrix structural decomposition, and subfield-linear design strategies. Contribution/Results: The framework yields a novel family of MSRD codes valid over arbitrary division rings—including finite and infinite fields. It unifies and generalizes prior MSRD constructions and classical Hamming-metric MDS codes. Over finite fields, it further produces new subfield-linear MDS codes with lengths approaching the field size, substantially expanding the existence range of optimal codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web search models and ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We extend the existing skew polynomial representations of matrix algebras which are direct sum of matrix spaces over division rings. In this representation, the sum-rank distance between two tuples of matrices is captured by a weight function on their associated skew polynomials, defined through degrees and greatest common right divisors with the polynomial that defines the representation. We exploit this representation to construct new families of maximum sum-rank distance (MSRD) codes over finite and infinite fields, and over division rings. These constructions generalize many of the known existing constructions of MSRD codes as well as of optimal codes in the rank and in the Hamming metric. As a byproduct, in the case of finite fields we obtain new families of MDS codes which are linear over a subfield and whose length is close to the field size.

Problem

• Extend skew polynomial representations for matrix algebras
• Construct new families of maximum sum-rank distance codes
• Generalize existing MSRD, rank, and Hamming metric codes

Innovation

• Extends skew polynomial representations of matrix algebras
• Constructs new maximum sum-rank distance codes
• Generalizes existing MSRD and optimal code constructions