A Lagrange–Dual Lower Bound to the Error Exponent of the Typical Random Code

A Lagrange-dual (Gallager-style) lower bound is derived for the error exponent function of the typical random code (TRC) pertaining to the i.i.d. random coding ensemble and mismatched stochastic likelihood decoding. While the original expression, derived from the method of types (the Csiszár-style expression) involves minimization over probability distributions defined on the channel input-output alphabets, the new Lagrange-dual formula involves optimization of five parameters, independently of the alphabet sizes. For both stochastic and deterministic mismatched decoding (including maximum likelihood decoding as a special case), we provide a rather comprehensive discussion on the insight behind the various ingredients of this formula and describe how its behavior varies as the coding rate exhausts the relevant range. Among other things, it is demonstrated that this expression simultaneously generalizes both the expurgated error exponent function (at zero rate) and the classical random coding exponent function at high rates, where it also meets the sphere-packing bound.