Independent parallel q-ary symmetric channels are a suitable transmission model for several applications. The proposed weighted-Hamming metric is tailored to this setting and enables optimal decoding performance. We show that some weighted-Hamming-metric codes exhibit the unusual property that all errors beyond half the minimum distance can be corrected. Nevertheless, a tight relation between the error-correction capability of a code and its minimum distance can be established. Generalizing their Hamming-metric counterparts, upper and lower bounds on the cardinality of a code with a given weighted-Hamming distance are obtained. Finally, we propose a simple code construction with optimal minimum distance for specific parameters.

翻译：独立并行q元对称信道是多种应用场景的合适传输模型。本文提出的加权汉明度量针对该场景进行了定制，并实现了最优解码性能。我们证明，某些加权汉明度量编码具有异常特性，即能够纠正超过最小距离一半的所有错误。尽管如此，编码的纠错能力与其最小距离之间仍可建立紧密关系。通过推广汉明度量对应的结论，我们获得了给定加权汉明距离下编码基数（cardinality）的上下界。最后，针对特定参数，我们提出了一种具有最优最小距离的简单编码构造方案。