# Exponential Source/Channel Duality

## TL;DR

This paper establishes a duality between source and channel coding in the exponential regime, deriving exact exponents for various coding scenarios and showing their interrelations through a unified framework.

## Contribution

It introduces a novel source/channel duality in the exponential regime and derives exact exponents for multiple coding schemes, unifying source and channel coding error analysis.

## Key findings

- Exact exponents for lossy source coding success and failure.
- Unified derivation of channel decoding error exponents from source coding exponents.
- Validation of Forney's bounds and derivation of the optimal erasure/list decoding exponent.

## Abstract

We propose a source/channel duality in the exponential regime, where success/failure in source coding parallels error/correctness in channel coding, and a distortion constraint becomes a log-likelihood ratio (LLR) threshold. We establish this duality by first deriving exact exponents for lossy coding of a memoryless source P, at distortion D, for a general i.i.d. codebook distribution Q, for both encoding success (R < R(P,Q,D)) and failure (R > R(P,Q,D)). We then turn to maximum likelihood (ML) decoding over a memoryless channel P with an i.i.d. input Q, and show that if we substitute P=QP, Q=Q, and D=0 under the LLR distortion measure, then the exact exponents for decoding-error (R < I(Q, P)) and strict correct-decoding (R > I(Q, P)) follow as special cases of the exponents for source encoding success/failure, respectively. Moreover, by letting the threshold D take general values, the exact random-coding exponents for erasure (D > 0) and list decoding (D < 0) under the simplified Forney decoder are obtained. Finally, we derive the exact random-coding exponent for Forney's optimum tradeoff erasure/list decoder, and show that at the erasure regime it coincides with Forney's lower bound and with the simplified decoder exponent.