Typical-set indexing example ($L_n=\lceil {\log }_2(|T_{\epsilon }^{(n)}|+1)\rceil$)

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 usesbits and has reconstruction-error probability at most $1-P(T_{\epsilon }^{(n)})$.

For this example and $\epsilon =0.1$, $n=40$ 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.