We present an approach to showing that a linear code is resilient to random errors. We then use this approach to obtain decoding results for both transitive codes and Reed-Muller codes. We give three kinds of results about linear codes in general, and transitive linear codes in particular. 1) We give a tight bound on the weight distribution of every transitive linear code $C \subseteq \mathbb{F}_2^N$: $\Pr_{c \in C}[|c| = \alpha N] \leq 2^{-(1-h(\alpha)) \mathsf{dim}(C)}$. 2) We give a Fourier analytic criterion that certifies that a linear code $C$ can be decoded on the binary symmetric channel. Let $L_w(x)$ denote the level function that is $1$ if $|x| =w$ and $0$ otherwise, and let $C^\perp$ denote the dual code of $C$. We show that bounds on $\mathbb{E}_{c \in C^{\perp}}[ \hat{L}_{\epsilon N}(c)^2]$ imply that $C$ recovers from errors on the binary symmetric channel with parameter $\epsilon$. Weaker bounds can be used to obtain list-decoding results using similar methods. 3) Motivated by the above results, we use complex analysis to give tight estimates for the Fourier coefficients of the level function. We then combine these estimates with our weight distribution bound to give list-decoding results for transitive linear codes and Reed-Muller codes.

翻译：我们提出了一个方法来显示线性代码对随机错误的适应性。 然后我们用这个方法来获取中转代码和Reed- Muller代码的解码结果。 我们给出了三种关于一般线性代码的结果, 特别是中转线性代码的结果。 1) 我们给每个中转线代码的重量分布设定了严格的限制 $C\subseteq \\ mathb{F\\\\\\\\\\\\\\\N$美元 美元: $\\\\\\\\\\\\\\\\\\\\\ =\ alphaN} 。 我们给出了一个Fourier 分析标准, 确定线性代码$C$C\\\\\\\\\\\\\\ 美元可以在双倍的代码中解码中解码。 如果 $xxxxx===w美元, 和$0xxxxxxxxx, 然后让 美元从 \\\\\\\\\\\\\\\xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx