# Error Exponents of Typical Random Codes for the Colored Gaussian Channel

## TL;DR

This paper characterizes the error exponents of typical random codes for the colored Gaussian channel, especially at low rates, highlighting differences from traditional exponents and providing tight formulas for matched and mismatched decoding.

## Contribution

It introduces a new definition of error exponent for typical random codes and derives explicit formulas for the colored Gaussian channel, including low-rate regimes and mismatched decoding scenarios.

## Key findings

- Error exponent formula is tight at low rates.
- Distinct error exponents for typical and average codes.
- Optimal input spectrum follows water pouring strategy.

## Abstract

The error exponent of the typical random code is defined as the asymptotic normalized expectation of the logarithm of the probability of error, as opposed to the traditional definition of the random coding exponent as the normalized logarithm of the expectation of the probability of error with respect to a given ensemble of codes. For a certain ensemble of independent codewords, with a given power spectrum, and a generalized stochastic mismatched decoder, we characterize the error exponent the typical random codes (TRC) for the colored Gaussian channel, with emphasis on the range of low rates, where the TRC error exponent differs in value from the ordinary random coding error exponent. The error exponent formula, which is exponentially tight at some range of low rates, is presented as the maximum of a certain function with respect to one parameter only (in the spirit of Gallager's formulas) in the case of matched decoding, and two parameters in the case of mismatched decoding. Several aspects of the main results are discussed. These include: general properties, a parametric representation, critical rates, phase transitions, optimal input spectrum (water pouring), and comparison to the random coding exponent.