---
title: Capacity of a Class of Deterministic Relay Channels
url: https://www.emergentmind.com/papers/0611053
type: paper
arxiv_id: '0611053'
arxiv_url: https://arxiv.org/abs/0611053
published: '2006-11-13'
categories:
- cs.IT
- math.IT
---

# Capacity of a Class of Deterministic Relay Channels

## Abstract

The capacity of a class of deterministic relay channels with the transmitter input X, the receiver output Y, the relay output Y_1 = f(X, Y), and a separate communication link from the relay to the receiver with capacity R_0, is shown to be C(R_0) = \max_{p(x)} \min \{I(X;Y)+R_0, I(X;Y, Y_1) \}. Thus every bit from the relay is worth exactly one bit to the receiver. Two alternative coding schemes are presented that achieve this capacity. The first scheme, ``hash-and-forward'', is based on a simple yet novel use of random binning on the space of relay outputs, while the second scheme uses the usual ``compress-and-forward''. In fact, these two schemes can be combined together to give a class of optimal coding schemes. As a corollary, this relay capacity result confirms a conjecture by Ahlswede and Han on the capacity of a channel with rate-limited state information at the decoder in the special case when the channel state is recoverable from the channel input and the output.