---
title: Error Exponents for Variable-length Block Codes with Feedback and Cost Constraints
url: https://www.emergentmind.com/papers/0612097
type: paper
arxiv_id: '0612097'
arxiv_url: https://arxiv.org/abs/0612097
published: '2006-12-20'
categories:
- cs.IT
- math.IT
---

# Error Exponents for Variable-length Block Codes with Feedback and Cost Constraints

## Abstract

Variable-length block-coding schemes are investigated for discrete memoryless channels with ideal feedback under cost constraints. Upper and lower bounds are found for the minimum achievable probability of decoding error $P_{e,\min}$ as a function of constraints $R, \AV$, and $\bar \tau$ on the transmission rate, average cost, and average block length respectively. For given $R$ and $\AV$, the lower and upper bounds to the exponent $-(\ln P_{e,\min})/\bar \tau$ are asymptotically equal as $\bar \tau \to \infty$. The resulting reliability function, $\lim_{\bar \tau\to \infty} (-\ln P_{e,\min})/\bar \tau$, as a function of $R$ and $\AV$, is concave in the pair $(R, \AV)$ and generalizes the linear reliability function of Burnashev to include cost constraints. The results are generalized to a class of discrete-time memoryless channels with arbitrary alphabets, including additive Gaussian noise channels with amplitude and power constraints.