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.

翻译：本文研究配备曼海姆距离的高斯整数上的线性码。我们发展了若干经典界限的曼海姆度量类比。推导出曼海姆球体积的显式公式，由此得到球包界以及两位纠错完美码参数的约束条件。我们证明了其他若干有用界限，并展示了对于某些参数满足这些界限的码族，从而表明这些界限是紧的。我们还讨论了高斯整数上的自对偶码，并使用麦克威廉姆斯型恒等式的曼海姆版本，获得了特定参数区域内其最小曼海姆距离的上界。最后，我们提出了高斯整数剩余类环上的译码算法。给出的实例表明，某些在汉明度量下不可纠正的错误在曼海姆度量下变为可纠正。