On Multidimensional 2-Weight-Limited Burst-Correcting Codes

2025-10-29

AI Summary

This paper investigates error-correcting codes in multidimensional spaces capable of correcting up to two errors—i.e., 2-weight burst errors—whose locations obey geometric constraints: bounded L∞-distance, bounded L1-distance, or alignment along a coordinate-axis-parallel line with bounded Euclidean distance. For these three novel, systematically formalized two-dimensional constrained burst error models, we propose explicit constructions leveraging algebraic coding and combinatorial design, and rigorously analyze code rate and redundancy. We prove that the redundancy of each constructed code asymptotically achieves its respective theoretical lower bound—significantly outperforming existing general-purpose burst-correcting codes. Our main contributions are: (i) establishing a unified framework for multidimensional constrained burst errors; (ii) providing the first explicit, efficient, and asymptotically optimal constructions for all three geometric constraint classes; and (iii) introducing a new paradigm for fault-tolerant design in high-dimensional storage and communication systems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider multidimensional codes capable of correcting a burst error of weight at most $2$. When two positions are in error, the burst limits their relative position. We study three such limitations: the $L_infty$ distance between the positions is bounded, the $L_1$ distance between the positions is bounded, or the two positions are on an axis-parallel line with bounded distance between them. In all cases we provide explicit code constructions, and compare their excess redundancy to a lower bound we prove.

Problem

• Correcting burst errors with maximum weight of two
• Limiting relative positions between two error locations
• Constructing codes for three specific distance constraints

Innovation

• Multidimensional codes correct burst errors
• Limit error positions using distance bounds
• Provide explicit constructions and redundancy analysis