Expurgated Random-Coding Ensembles: Exponents, Refinements and Connections

Abstract: 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.