The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by R{\'e}nyi divergence of order $\alpha \in [1,\infty]$. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner's transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed-Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel.

翻译：噪声算子作用于码字后，会将其转化为对应空间上的分布。信息论中的典型例子包括：伯努利噪声作用于汉明空间上的码字，以及高斯噪声作用于欧几里得空间上的格点。本文旨在刻画输出分布接近空间上均匀分布的条件——该接近程度通过阶数为 $\alpha \in [1,\infty]$ 的Rényi散度度量。该问题的某一版本在信息论中被称为信道可解性难题，其对窃听信道安全性保障、纠错编码、差异性问题、最坏情况到平均情况的复杂度归约等诸多问题具有重要启示。本研究量化了渐近均匀性（完美平滑化）所需的条件，并确定了在伯努利和球噪声算子作用下实现这一特性的显式码族。我们推导了实现渐近完美平滑化所需的最小码率表达式。在证明过程中，本文借助了汉明空间上函数调和分析的最新成果。另一项成果涉及码族在二元窃听信道中Wyner传输方案的应用：我们识别出能够在该方案中保证强安全性的显式码族，证明嵌套里德-穆勒码能以正速率在二元对称窃听信道上实现可靠且安全的信息传输。最后，本文建立了二元对称信道中平滑化与纠错编码之间的关联。