# Error Exponents of Typical Random Trellis Codes

## TL;DR

This paper derives error exponents for typical random trellis codes over discrete memoryless channels, providing formulas that help identify good codes and error events, and extends results to channels with memory and mismatch.

## Contribution

It introduces a Csiszar-style and Gallager-style error exponent analysis for typical random trellis codes, extending previous work on block codes to structured trellis codes.

## Key findings

- Derived a Csiszar-style error exponent formula for trellis codes.
- Established a Gallager-style error exponent related to the expurgated exponent.
- Extended analysis to channels with memory and mismatch.

## Abstract

In continuation to an earlier work, where error exponents of typical random codes were studied in the context of general block coding, with no underlying structure, here we carry out a parallel study on typical random, time-varying trellis codes for general discrete memoryless channels, focusing on a certain range of low rates. By analyzing an upper bound to the error probability of the typical random trellis code, using the method of types, we first derive a Csiszar-style error exponent formula (with respect to the constraint length), which allows to easily identify and characterize properties of good codes and dominant error events. We also derive a Gallager-style form of this error exponent, which turns out to be related to the expurgated error exponent. The main result is further extended to channels with memory and mismatch.