Perfect codes in weakly metric association schemes

2026-01-19

AI Summary

This study investigates the existence of perfect codes under non-classical metrics, including the Lee metric, the NRT metric, mixed Hamming metrics, and rank distance. By introducing a novel framework termed “polynomial weak-metric association schemes” and integrating tools such as Lloyd’s theorem, the Schwartz–Zippel lemma, and asymptotic counting methods from integer partitions, the work establishes deep connections between this framework and multivariate P-polynomial association schemes as well as m-distance-regular graph theory. Building on these connections, the authors prove the nonexistence of perfect codes in the aforementioned metrics, thereby substantially extending the known boundaries of coding theory in generalized metric spaces.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric association schemes. A connection between this notion and the recent theory of multivariate P-polynomial schemes of ( Bannai et al. 2025) and of $m$-distance regular graphs ( Bernard et al 2025) is pointed out.

Problem

• perfect codes
• Lee metric
• NRT metric
• sum-rank distance
• mixed Hamming metric

Innovation

• polynomial weakly metric association schemes
• perfect codes
• Schwartz-Zippel Lemma
• asymptotic enumeration
• sum-rank distance