  Using techniques and results from Kudekar et al. we strengthen the bounds on
the weight distribution of linear codes achieving capacity on the BEC, which
were shown by the first author. In particular, we show that for any doubly
transitive binary linear code $C \subseteq \{0,1\}^n$ of rate $0 < R < 1$ with
weight distribution $\left(a_0,...,a_n\right)$ holds $a_i \le 2^{o(n)} \cdot
(1-R)^{-2 \ln 2 \cdot \min\{i, n-i\}}$.
  For doubly transitive codes with minimal distance at least
$\Omega\left(n^c\right)$, $0 < c \le 1$, the error factor of $2^{o(n)}$ in this
bound can be removed at the cost of replacing $1-R$ with a smaller constant $a
= a(R,c) < 1- R$. Moreover, in the special case of Reed-Muller codes, due to
the additional symmetries of these codes, this error factor can be removed at
essentially no cost.
  This implies that for any doubly transitive code $C$ of rate $R$ with minimal
distance at least $\Omega\left(n^c\right)$, there exists a positive constant $p
= p(R,c)$ such that $C$ decodes errors on $\mathrm{BSC}(p)$ with high
probability if $p < p(R,c)$. For doubly transitive codes of a sufficiently low
rate (smaller than some absolute constant) the requirement on the minimal
distance can be omitted, and hence this critical probability $p(R)$ depends
only on $R$. Furthermore, $p(R) \rightarrow \frac12$ as $R \rightarrow 0$.
  In particular, a Reed-Muller code $C$ of rate $R$ decodes errors on
$\mathrm{BSC}(p)$ with high probability if \[ R ~<~ 1 -
\big(4p(1-p)\big)^{\frac{1}{4 \ln 2}}, \] answering a question of Abbe, Hazla,
and Nachum.
