We prove equivalence of using the modulus metric and Euclidean metric in solving the soft decoding problem for a memoryless discrete channel with binary input and Q-ary output. For such a channel, we give an example of a construction of binary codes correcting t binary errors in the Hamming metric. The constructed codes correct errors at the output of a demodulator with Q quantization errors as (t + 1)(Q − 1) − 1 errors in the modulus metric. The obtained codes are shown to have polynomial decoding complexity.

中文翻译：

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关于模量度量在求解最小欧氏距离译码问题​​中的应用

我们证明使用模量度量和欧几里德度量在解决软解码问题与二进制输入和一个无记忆离散信道的等效Q进制输出。对于这样的信道，我们给出了一个二进制代码构造的示例，该二进制代码可以纠正汉明度量中的t个二进制错误。所构造的代码用Q量化误差校正了解调器输出处的误差，该量化误差为模量度量中的（t +1）（Q -1）-1个误差。所获得的代码被示为具有多项式解码复杂度。