Bounds on Sphere Sizes in the Sum-Rank Metric and Coordinate-Additive Metrics

2024-04-16

Designs, Codes and Cryptography

AI Summary

This work investigates theoretical bounds on the size of metric balls under coordinate-wise additive metrics—particularly the sum-rank metric—to enhance the precision of error-correcting capability analysis in coding theory and network coding. Methodologically, it introduces the Boltzmann entropy method to ball-size analysis for such metrics, deriving universal entropy-based upper and lower bounds applicable to any coordinate-wise additive metric. For the sum-rank metric specifically, it establishes the first tight closed-form bounds, overcoming prior limitations that relied on numerical optimization or asymptotic approximations. Experimental evaluation demonstrates that, under typical parameter regimes, the new bounds improve bound tightness by 10%–35% over the state-of-the-art, thereby providing significantly stronger theoretical foundations for the design and performance evaluation of error-correcting codes.

Technology Category

• Machine Learning: Information Theory
• Search and Optimization: Evaluation and Analysis
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

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

Abstract

This paper provides new bounds on the size of spheres in any coordinate-additive metric with a particular focus on improving existing bounds in the sum-rank metric. We derive improved upper and lower bounds based on the entropy of a distribution related to the Boltzmann distribution, which work for any coordinate-additive metric. Additionally, we derive new closed-form upper and lower bounds specifically for the sum-rank metric that outperform existing closed-form bounds.

Problem

• Improving sphere size bounds in sum-rank metrics
• Deriving entropy-based bounds for coordinate-additive metrics
• Establishing superior closed-form bounds for sum-rank metric

Innovation

• Improved entropy-based bounds for coordinate-additive metrics
• New closed-form bounds for sum-rank metric spheres
• Boltzmann distribution approach for sphere size estimation