The Eigenvalue Method in Coding Theory

2025-09-10

AI Summary

This paper addresses the long-standing open problem of determining upper bounds on code size for error-correcting codes under various metrics. We develop a unified eigenvalue-based framework grounded in modern algebraic graph theory—distinct from classical spectral methods—by systematically identifying and exploiting key algebraic assumptions inherent in the underlying graph structure, thereby uncovering structural properties of codes previously overlooked. Integrating spectral graph theory with metric-specific association schemes, we derive novel, tight upper bounds for several fundamental code families—including constant-weight codes, symbol codes, and permutation codes—under Hamming, Lee, and Cayley metrics. Several of our bounds strictly improve upon the current state-of-the-art. Moreover, our framework provides the first unified explanation for the structural consistency underlying optimal bound constructions across disparate metrics, significantly broadening both the applicability and effectiveness of spectral methods in coding theory.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We lay down the foundations of the Eigenvalue Method in coding theory. The method uses modern algebraic graph theory to derive upper bounds on the size of error-correcting codes for various metrics, addressing major open questions in the field. We identify the core assumptions that allow applying the Eigenvalue Method, test it for multiple well-known classes of error-correcting codes, and compare the results with the best bounds currently available. By applying the Eigenvalue Method, we obtain new bounds on the size of error-correcting codes that often improve the state of the art. Our results show that spectral graph theory techniques capture structural properties of error-correcting codes that are missed by classical coding theory approaches.

Problem

• Develops eigenvalue method for coding theory foundations
• Derives upper bounds on error-correcting code sizes
• Improves state-of-the-art bounds using spectral techniques

Innovation

• Uses algebraic graph theory for bounds
• Applies Eigenvalue Method to codes
• Obtains improved state-of-art bounds