Gallager, random-coding, expurgated, and sphere-packing functions

Let $W$ be a finite unconstrained DMC from finite nonempty $\mathcal{X}$ to $\mathcal{Y}$, with rates and logarithms in nats. The one-letter function $E_0(\rho ,P)$ for $\rho \ge 0$ and $E_{0x}(\rho ,P)$ for $\rho \ge 1$ are defined in One-letter Gallager and expurgated exponent functions. Their maxima over the input simplex are attained and finite by Gallager and expurgated input-law maxima are attained. For every $\rho \ge 0$, define $E_0(\rho ):={\max }_P{E}_0(\rho ,P)$; on the random-coding range set $E_r(R):={\sup }_{0\le \rho \le 1}\{E_0(\rho )-\rho R\}\mathrm{.}$ The expurgated function and sphere-packing expression are $E_{\mathrm{e}\mathrm{x}}(R):={\sup }_{\rho \ge 1}\{{\max }_P{E}_{0x}(\rho ,P)-\rho R\},E_{\mathrm{s}\mathrm{p}}(R):={\sup }_{\rho \ge 0}\{E_0(\rho )-\rho R\}\mathrm{.}$ These functions define lower or upper bounds used on this page; they do not assert that an error-exponent limit exists or that the sphere-packing bound applies without its later strict-feasibility hypothesis.