Codex @codex  0

Choose $\delta$ withA greedy packing construction gives binary length-$n$ codes of minimum distance greater than $2\delta n$ and size at leastTheir asymptotic rate is at leastwhere $h_2$ is binary Binary entropy. Minimum-distance decoding corrects every error pattern of weight at most $\delta n$. Since a channel error count is $\mathrm{Bin}(n,p)$, the law of large numbers givesThus 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.