Code smoothing is a phenomenon in which an error distribution makes a code statistically close to the uniform distribution over the ambient space. This closeness is measured by total variation distance. Recently, Debris-Alazard et al.\ introduced a smoothing bound, which is an upper bound on this total variation distance. Although the smoothing bound evaluates how the error distribution smooths a code, this bound applies only to linear codes. In this paper, we generalize this bound to not only linear codes but also specific non-linear codes. While the smoothing bound in previous work was obtained by Fourier analysis over finite abelian groups, we derive this bound using a graph-theoretic approach. To derive the smoothing bound, we consider code smoothing as the mixing of random walks on a specific graph, and use the concept of equitable partitions, which is well-studied in graph theory.

翻译：编码平滑是一种现象，其中误差分布使得编码在统计上接近于环境空间上的均匀分布。这种接近度通过总变差距离来衡量。最近，Debris-Alazard等人提出了一种平滑界，即该总变差距离的上界。尽管平滑界评估了误差分布如何平滑编码，但该界仅适用于线性编码。在本文中，我们将该界推广到不仅适用于线性编码，还适用于特定的非线性编码。虽然先前工作中的平滑界是通过对有限阿贝尔群进行傅里叶分析得到的，但我们采用图论方法推导了该界。为推导平滑界，我们将编码平滑视为特定图上随机游走的混合过程，并利用了图论中研究成熟的公平划分概念。