Weighted-Hamming Metric: Bounds and Codes

2026-01-19

AI Summary

This work addresses the overly conservative nature of traditional minimum-distance-based estimates of error-correcting capability under the weighted Hamming metric. Breaking through this limitation, the paper establishes, for the first time, a tighter bound on the direct error-correcting capability and introduces an efficient constructible code based on a generalized concatenated structure. Coupled with a tailored decoding algorithm, the proposed method enables reliable decoding beyond the conventional radius of half the minimum distance, significantly enhancing practical error-correction performance while maintaining low computational complexity.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds

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

The weighted-Hamming metric generalizes the Hamming metric by assigning different weights to blocks of coordinates. It is well-suited for applications such as coding over independent parallel channels, each of which has a different level of importance or noise. From a coding-theoretic perspective, the actual error-correction capability of a code under this metric can exceed half its minimum distance. In this work, we establish direct bounds on this capability, tightening those obtained via minimum-distance arguments. We also propose a flexible code construction based on generalized concatenation and show that these codes can be efficiently decoded up to a lower bound on the error-correction capability.

Problem

• weighted-Hamming metric
• error-correction capability
• minimum distance
• coding theory
• bounds

Innovation

• weighted-Hamming metric
• error-correction capability
• generalized concatenation
• code construction
• decoding bounds