 Solution

For an IID finite-alphabet source $Q$ and a fixed rate $H(Q)<R<{\log }_2|\mathcal{A}|$, let $P_e^{\ast }(n,R)$ be the smallest probability that a block is not represented by a fixed-to-fixed code having at most $2^{nR}$ codewords. The fixed-rate source-coding error exponent theorem states

For the direct part, encode all type classes whose empirical entropy is at most $R-{\delta }_n$, where ${\delta }_n\downarrow 0$ and $n{\delta }_n-|\mathcal{A}|{\log }_2(n+1)\to \infty$. Since a type class has at most $2^{nH(P)}$ sequences and there are at most $(n+1{)}^{|\mathcal{A}|}$ types, this codebook has at most $2^{nR}$ entries for large $n$. An error can occur only when $H({\hat{P}}_n)>R-{\delta }_n$. The method of types bounds its probability byTaking the lower limit of the exponent and using compactness and continuity gives at least $D^{\ast }(R,Q)$, proving achievability.