# A Lagrange-Dual Lower Bound to the Error Exponent Function of the Typical Random Code

## TL;DR

This paper introduces a Lagrange-dual lower bound for the error exponent of the typical random code, simplifying optimization and unifying various error exponent bounds in information theory.

## Contribution

It derives a novel Lagrange-dual formula for the error exponent, reducing complexity and unifying bounds for mismatched decoding scenarios.

## Key findings

- The new formula involves optimization over five parameters, independent of alphabet sizes.
- It generalizes both the expurgated and random coding error exponents.
- The expression meets the sphere-packing bound at high rates.

## Abstract

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 Csiszar-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.