This article studies the efficacy of the Metropolis algorithm for the minimum-weight codeword problem. The input is a linear code $C$ given by its generator matrix and our task is to compute a nonzero codeword in the code $C$ of least weight. In particular, we study the Metropolis algorithm on two possible search spaces for the problem: 1) the codeword space and 2) the generator space. The former is the space of all codewords of the input code and is the most natural one to use and hence has been used in previous work on this problem. The latter is the space of all generator matrices of the input code and is studied for the first time in this article. In this article, we show that for an appropriately chosen temperature parameter the Metropolis algorithm mixes rapidly when either of the search spaces mentioned above are used. Experimentally, we demonstrate that the Metropolis algorithm performs favorably when compared to previous attempts. When using the generator space, the Metropolis algorithm is able to outperform the previous algorithms in most of the cases. We have also provided both theoretical and experimental justification to show why the generator space is a worthwhile search space to use for this problem.

中文翻译：

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使用码字和生成器搜索空间解决最小权重码字问题的 Metropolis 算法的有效性

本文研究了 Metropolis 算法对 最小权码字问题. 输入是一个线性代码 $C$ 由其生成矩阵给出，我们的任务是计算代码中的非零码字 $C$ 最轻的。特别是，我们在两个可能的搜索空间上研究了 Metropolis 算法：1）码字空间 和 2） 发电机空间. 前者是输入代码的所有码字的空间，是最自然使用的空间，因此在之前的工作中已经使用过这个问题。后者是输入代码的所有生成矩阵的空间，本文第一次研究。In this article, we show that for an appropriately chosen temperature parameter the Metropolis algorithm mixes rapidly when either of the search spaces mentioned above are used. 通过实验，我们证明了与之前的尝试相比，Metropolis 算法的性能更好。当使用生成器空间时，Metropolis 算法在大多数情况下能够胜过以前的算法。我们还提供了理论和实验证明来说明为什么生成器空间是一个值得用于这个问题的搜索空间。