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

TL;DR

This paper introduces new bounds and theoretical results for linear codes over Gaussian integers using the Mannheim distance, including sphere packing bounds, bounds on self-dual codes, and decoding algorithms, demonstrating improved error correction capabilities.

Contribution

It develops Mannheim-metric analogues of classical bounds, derives explicit volume formulas, and presents tight bounds and decoding methods for codes over Gaussian integers.

Findings

Derived explicit volume formula for Mannheim balls

Established sphere packing and other bounds for Mannheim codes

Presented decoding algorithms and examples of improved error correction

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.