New bounds for codes over Gaussian integers based on the Mannheim distance
This work addresses incremental improvements in coding theory for specialized applications, focusing on theoretical bounds and algorithms for Gaussian integer codes.
The paper tackles the problem of analyzing linear codes over Gaussian integers using the Mannheim distance, deriving new bounds and decoding algorithms, with results including explicit formulas for ball volumes and examples showing improved error correction compared to the Hamming metric.
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.