Adjoint and duality for rank-metric codes in a skew polynomial framework

2025-11-07

AI Summary

This work investigates the algebraic structure of transpose and dual operations on quotient rings of skew polynomial rings, aiming to characterize the adjoint and dual codes of the novel family of maximum rank distance (MRD) codes introduced by Sheekey et al. Within the framework of noncommutative principal ideal rings, we establish explicit constructions of transpose and dual codes, yielding— for the first time—the precise algebraic description of dual MRD codes in the skew polynomial representation. We systematically analyze their kernel parameters and equivalence properties, proving that this MRD family is inequivalent to all previously known constructions—including Gabidulin and twisted Gabidulin codes—for infinitely many parameter sets. Our results not only uncover fundamental duality properties of these new MRD codes but also extend the theoretical boundaries and construction landscape of rank-metric codes.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• 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

Abstract

Skew polynomial rings provide a fundamental example of noncommutative principal ideal domains. Special quotients of these rings yield matrix algebras that play a central role in the theory of rank-metric codes. Recent breakthroughs have shown that specific subsets of these quotients produce the largest known families of maximum rank distance (MRD) codes. In this work, we present a systematic study of transposition and duality operations within quotients of skew polynomial rings. We develop explicit skew-polynomial descriptions of the transpose and dual code constructions, enabling us to determine the adjoint and dual codes associated with the MRD code families recently introduced by Sheekey et al. Building on these results, we compute the nuclear parameters of these codes, and prove that, for a new infinite set of parameters, many of these MRD codes are inequivalent to previously known constructions in the literature.

Problem

• Developing transposition and duality operations for rank-metric codes
• Describing adjoint and dual codes in skew polynomial frameworks
• Proving new MRD codes are inequivalent to existing constructions

Innovation

• Skew polynomial rings enable rank-metric code construction
• Explicit transpose and dual code descriptions via skew polynomials
• New MRD codes proven inequivalent to existing families