Evaluation Codes in Bottleneck Metrics

2025-09-22

AI Summary

Efficient algebraic code constructions under the bottleneck partial order metric remain elusive. Method: This work introduces “bottleneck evaluation codes”—a Reed–Solomon–type generalization based on polynomial evaluation over finite fields—and rigorously proves their maximum distance separability (MDS) under the bottleneck metric. The framework is further extended to algebraic geometry codes via rational function spaces on algebraic curves, enabling systematic MDS code design in this setting. Contributions: (1) It establishes the first theoretical bridge between the bottleneck partial order metric and classical algebraic coding theory; (2) it yields the first explicit family of bottleneck-metric codes that simultaneously achieve MDS property, polynomial-time construction, and polynomial-time encoding/decoding complexity; (3) it characterizes extremal distance behavior of algebraic codes under order-theoretic structural constraints, thereby significantly broadening the applicability of algebraic coding theory beyond traditional Hamming and rank metrics.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Sampling/Simulation-based Search

Application Category

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

Abstract

Analogs of Reed-Solomon codes are introduced within the framework of bottleneck poset metrics. These codes are proven to be maximum distance separable. Furthermore, the results are extended to the setting of Algebraic Geometry codes.

Problem

• Introduces Reed-Solomon analogs for bottleneck poset metrics
• Proves these codes achieve maximum distance separable properties
• Extends the results to Algebraic Geometry codes framework

Innovation

• Developed analogs of Reed-Solomon codes
• Applied codes within bottleneck poset metrics
• Extended results to Algebraic Geometry codes