# Variable-length codes for channels with memory and feedback: error-exponent lower bounds

## TL;DR

This paper establishes lower bounds on the error-exponent for variable-length coding over unifilar channels with feedback, using a two-stage scheme and analyzing the log-likelihood ratio drift, supported by simulations.

## Contribution

It introduces a novel two-stage transmission scheme for unifilar channels with feedback and derives lower bounds on the reliability function, extending classical results to channels with memory.

## Key findings

- Lower bounds on error-exponent for unifilar channels with feedback.
- Two-stage scheme effectively improves reliability.
- Simulation results show bounds are tight.

## Abstract

The reliability function of memoryless channels with noiseless feedback and variable-length coding has been found to be a linear function of the average rate in the classic work of Burnashev. In this work we consider unifilar channels with noiseless feedback and study specific transmission schemes, the performance of which provides lower bounds for the channel reliability function. In unifilar channels the channel state evolves in a deterministic fashion based on the previous state, input, and output, and is known to the transmitter but is unknown to the receiver. We consider a two-stage transmission scheme. In the first stage, both transmitter and receiver summarize their common information in an M-dimensional vector with elements in the state space of the unifilar channel and an M-dimensional probability mass function, with M being the number of messages. The second stage, which is entered when one of the messages is sufficiently reliable, is resolving a binary hypothesis testing problem. The analysis assumes the presence of some common randomness shared by the transmitter and receiver, and is based on the study of the log-likelihood ratio of the transmitted message posterior belief, and in particular on the study of its multi-step drift. Simulation results confirm that the bounds are tight compared to the upper bounds derived in a companion paper.