Abstract: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 the 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.
From: Yusaku Nishimura [view email]
[v1]
Wed, 18 Mar 2026 06:56:21 UTC (9 KB)
[v2]
Mon, 8 Jun 2026 02:56:32 UTC (11 KB)
[v3]
Wed, 10 Jun 2026 05:40:50 UTC (11 KB)
[v4]
Fri, 4 Sep 2026 04:41:13 UTC (13 KB)
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