Average and maximal error reliability exponents

For a finite DMC $W$, $R>0$, and $n\ge 1$, set $M_n=\lceil e^{nR}\rceil$. Let $P_k^{\ast }(n,M_n)$ be the minimum error probability among length-$n$ deterministic codes of that size under criterion $k\in \{\mathrm{a}\mathrm{v},\mathrm{m}\mathrm{a}\mathrm{x}\}$, where the two errors are those of Average and maximal error of a fixed-length code. This minimum exists because the encoder and decoder range over finite sets. Define $E_k^{-}(R):={\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}-\frac{1}{n}\ln P_k^{\ast }(n,M_n),E_k^{+}(R):={\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}-\frac{1}{n}\ln P_k^{\ast }(n,M_n),$ with $-\ln 0=+\mathrm{\infty }$. An error-exponent limit $E_k(R)$ is used only when the liminf and limsup agree in the extended reals. These are exponents of error probability; the exponent of success probability is a different quantity. No general exact formula for the finite-DMC reliability function is part of this definition.