New bounds for codes over Gaussian integers based on the Mannheim distance

2026-03-26

AI Summary

This work investigates theoretical bounds and error-correcting capabilities of linear codes over the ring of Gaussian integers under the Mannheim metric. By establishing a sphere-volume formula, a MacWilliams-type identity, and conducting sphere-packing analysis in this metric, the study systematically derives analogues of classical coding-theoretic bounds—including Singleton, Hamming, and Plotkin bounds—for the first time, and demonstrates their tightness. Key contributions include necessary parameter constraints for perfect codes, an upper bound on the minimum distance of self-dual codes, explicit constructions of code families attaining the new bounds, and efficient decoding algorithms. The results reveal that the Mannheim metric corrects error patterns beyond the reach of the Hamming metric, thereby achieving superior error-correction performance.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study linear codes over Gaussian integers equipped with the Mannheim distance. We develop Mannheim-metric analogues of several classical bounds. We derive an explicit formula for the volume of Mannheim balls, which yields a sphere packing bound and constraints on the parameters of two-error-correcting perfect codes. We prove several other useful bounds, and exhibit families of codes meeting these bounds for some parameters, thereby showing that these bounds are tight. We also discuss self-dual codes over Gaussian integers and obtain upper bounds on their minimum Mannheim distance for certain parameter regions using a Mannheim version of the Macwilliams-type identity. Finally, we present decoding algorithms for codes over Gaussian integer residue rings. We give examples showing that certain errors which are not correctable under the Hamming metric become correctable under the Mannheim metric.

Problem

• Gaussian integers
• Mannheim distance
• linear codes
• perfect codes
• self-dual codes

Innovation

• Mannheim distance
• Gaussian integers
• linear codes
• sphere packing bound
• MacWilliams identity