= Typical-set indexing example
{title2=$L_n=\lceil\log_2(|T_\varepsilon^{(n)}|+1)\rceil$}
The <typical sequence theorem> turns a probability concentration into a coding construction. Assign one <binary number> index to each <weakly typical sequence>, plus a reserved index for every word outside the set. The <decoder> recovers every typical word exactly; the reserved index has one predetermined output. The <block code> uses
$$
L_n=\left\lceil\log_2(|T_\varepsilon^{(n)}|+1)\right\rceil
$$
<bits> and has reconstruction-error <probability> at most $1-P(T_\varepsilon^{(n)})$.
For this example and $\varepsilon=0.1$, $n=40$ needs 34 <bits> but permits error up to about $0.3611$. At $n=640$, 570 <bits> suffice and the error bound is about $0.0002215$. These are existence calculations for an explicitly enumerated collection, rather than claims about an efficient practical <encoder>. The reduction comes from protecting the whole high-probability collection, rather than just its single most likely member.
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