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：码平滑是一种现象，指误差分布使码字在环境空间上统计逼近均匀分布，其接近程度由全变差距离度量。近期，Debris-Alazard等人引入了一种平滑界，即该全变差距离的上界。尽管平滑界能够评估误差分布对码字的平滑效果，但该界仅适用于线性码。本文将该界推广至不仅涵盖线性码，还适用于特定非线性码。先前工作中的平滑界是通过有限阿贝尔群上的傅里叶分析获得的，而本文则采用图论方法推导该界。为推导平滑界，我们将码平滑视为特定图上随机游走的混合过程，并利用图论中已充分研究的可划分概念。