The noise distribution on $F_2^n$: probability of landing in a subspace versus a coset
The probability that $x \in C + e$ is equal to $2^n \langle T_\rho 1_{0}, 1_{C+e} \rangle$, where $T_\rho$ is the noise operator and $\rho = 1-2\epsilon&gt;0$. Now
$$
2^n \langle T_\rho 1_0, 1_{C+e} \rangle = 2^n \langle T_\rho 1_e, 1_C \rangle = \frac{2^n}{|C|} \langle T_\rho 1_{C+e}, 1_C \rangle.
$$
In terms of the Fourier expansion, this quantity equals
$$
\frac{2^n}{|C|} \sum_S \rho^{|S|} \widehat{1_C}(S) \widehat{1_{C+e}}(S) = \frac{2^n}{|C|} \sum_S \rho^{|S|} \widehat{1_C}(S)^2 \chi_S(e).
$$
Since $\rho &gt; 0$, this is maximized uniquely when $\chi_S(e) = 1$ whenever $\widehat{1_C}(S) \neq 0$. In other words, the quantity is maximized uniquely for $e$ satisfying $C + e = C$.

Note that even when $C$ isn't a code, we have shown that if you start at a random point in $C$, apply the noise, and measure the probability that you reach $C + e$, then this probability is maximized for $e$ satisfying $C + e = C$, and in particular for $e = 0$.