= Solution
Choose $\delta$ with
$$
p<\delta<\frac14.
$$
A greedy packing construction gives binary length-$n$ codes of minimum distance greater than $2\delta n$ and size at least
$$
\frac{2^n}{\sum_{j\leq2\delta n}\binom nj}.
$$
Their asymptotic rate is at least
$$
1-h_2(2\delta)>0,
$$
where $h_2$ is binary <Binary entropy>. Minimum-distance decoding corrects every error pattern of weight at most $\delta n$. Since a channel error count is $\operatorname{Bin}(n,p)$, the law of large numbers gives
$$
\mathbb P\{\operatorname{Bin}(n,p)>\delta n\}\longrightarrow0.
$$
\b[Thus a fixed positive rate is achievable with error tending to zero, proving that the operational capacity is nonzero. This is the <positive-rate coding bound below one-quarter crossover>.]
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