We give a quantum reduction from finding short codewords in a random linear code to decoding for the Hamming metric. This is the first time such a reduction (classical or quantum) has been obtained. Our reduction adapts to linear codes Stehl\'e-Steinfield-Tanaka-Xagawa' re-interpretation of Regev's quantum reduction from finding short lattice vectors to solving the Closest Vector Problem. The Hamming metric is a much coarser metric than the Euclidean metric and this adaptation has needed several new ingredients to make it work. For instance, in order to have a meaningful reduction it is necessary in the Hamming metric to choose a very large decoding radius and this needs in many cases to go beyond the radius where decoding is always unique. Another crucial step for the analysis of the reduction is the choice of the errors that are being fed to the decoding algorithm. For lattices, errors are usually sampled according to a Gaussian distribution. However, it turns out that the Bernoulli distribution (the analogue for codes of the Gaussian) is too much spread out and cannot be used, as such, for the reduction with codes. This problem was solved by using instead a truncated Bernoulli distribution.

翻译：[翻译后摘要] 本文给出了从随机线性码中寻找短码字到汉明度量译码问题的量子归约。这是此类归约（无论是经典还是量子）首次被实现。我们的归约借鉴了Stehlé–Steinfield–Tanaka–Xagawa对Regev量子归约（将短格向量寻找问题转化为最近向量问题）的重新诠释，并针对线性码进行了适配。汉明度量相比欧几里得度量更为粗糙，因此这种适配需要多个新要素才能实现。例如，为构建有意义的归约，必须在汉明度量下选择非常大的译码半径，而在许多情况下，这一半径需超出译码唯一性成立的界限。归约分析的另一个关键步骤是选择输入译码算法的错误模式。对于格，错误通常按高斯分布采样。然而，伯努利分布（高斯分布在编码理论中的对应物）过于分散，无法直接用于编码的归约。为此，我们改用截断伯努利分布解决了这一问题。