  This paper studies expurgated random-coding bounds and exponents for channel
coding with a given (possibly suboptimal) decoding rule. Variations of
Gallager's analysis are presented, yielding several asymptotic and
non-asymptotic bounds on the error probability for an arbitrary codeword
distribution. A simple non-asymptotic bound is shown to attain an exponent of
Csisz\'ar and K\"orner under constant-composition coding. Using Lagrange
duality, this exponent is expressed in several forms, one of which is shown to
permit a direct derivation via cost-constrained coding which extends to
infinite and continuous alphabets. The method of type class enumeration is
studied, and it is shown that this approach can yield improved exponents and
better tightness guarantees for some codeword distributions. A generalization
of this approach is shown to provide a multi-letter exponent which extends
immediately to channels with memory. Finally, a refined analysis expurgated
i.i.d. random coding is shown to yield a O\big(\frac{1}{\sqrt{n}}\big)
prefactor, thus improving on the standard O(1) prefactor. Moreover, the implied
constant is explicitly characterized.
