#
A New Coding Paradigm for the Primitive Relay Channel^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Existing Upper and Lower Bounds

#### 2.1. Cut-Set Upper Bound

#### 2.2. Improvements on a Cut-Set Upper Bound

- Symmetric (${Y}_{\mathrm{SR}}$ and ${Y}_{\mathrm{SD}}$ are conditionally identically distributed given ${X}_{\mathrm{S}}$): $I({X}_{\mathrm{S}};{Y}_{\mathrm{SD}},{\tilde{Y}}_{\mathrm{SR}})=I({X}_{\mathrm{S}};{Y}_{\mathrm{SD}})$.
- Degraded (${Y}_{\mathrm{SD}}$ is a stochastically degraded version of ${Y}_{\mathrm{SR}}$): $I({X}_{\mathrm{S}};{Y}_{\mathrm{SD}},{\tilde{Y}}_{\mathrm{SR}})=I({X}_{\mathrm{S}};{Y}_{\mathrm{SR}})$.
- Reversely degraded (${Y}_{\mathrm{SR}}$ is a stochastically degraded version of ${Y}_{\mathrm{SD}}$): $I({X}_{\mathrm{S}};{Y}_{\mathrm{SD}},{\tilde{Y}}_{\mathrm{SR}})=I({X}_{\mathrm{S}};{Y}_{\mathrm{SD}})$.

#### 2.3. Direct Transmission Lower Bound

- the primitive relay channel is reversely degraded, which implies that $I({X}_{\mathrm{S}};{Y}_{\mathrm{SD}})=I({X}_{\mathrm{S}};{Y}_{\mathrm{SR}},{Y}_{\mathrm{SD}})$;
- ${C}_{\mathrm{RD}}=0$.

#### 2.4. Decode-and-Forward Lower Bound

- the primitive relay channel is degraded, which implies that $I({X}_{\mathrm{S}};{Y}_{\mathrm{SR}})=I({X}_{\mathrm{S}};{Y}_{\mathrm{SR}},{Y}_{\mathrm{SD}})$;
- $I({X}_{\mathrm{S}};{Y}_{\mathrm{SR}})\ge I({X}_{\mathrm{S}};{Y}_{\mathrm{SD}})+{C}_{\mathrm{RD}}$.

#### 2.5. Partial Decode-and-Forward Lower Bound

#### 2.6. Compress-and-Forward Lower Bound

#### 2.7. Partial Decode-Compress-and-Forward Lower Bound

## 3. Main Result

**Theorem**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Corollary**

**1.**

**Proof**

**of**

**Theorem**

**1**

**.**

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**The erasure relay channel: primitive relay channel in which the link from source to relay is a BEC$\left({\epsilon}_{\mathrm{SR}}\right)$ and the link from source to destination is a BEC$\left({\epsilon}_{\mathrm{SD}}\right)$.

**Figure 4.**Comparison between the achievable rate provided by our strategy and the existing upper and lower bounds. We use “CF” and “DF” as abbreviations for “compress-and-forward” and “decode-and-forward”, respectively.

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**MDPI and ACS Style**

Mondelli, M.; Hassani, S.H.; Urbanke, R.
A New Coding Paradigm for the Primitive Relay Channel. *Algorithms* **2019**, *12*, 218.
https://doi.org/10.3390/a12100218

**AMA Style**

Mondelli M, Hassani SH, Urbanke R.
A New Coding Paradigm for the Primitive Relay Channel. *Algorithms*. 2019; 12(10):218.
https://doi.org/10.3390/a12100218

**Chicago/Turabian Style**

Mondelli, Marco, S. Hamed Hassani, and Rüdiger Urbanke.
2019. "A New Coding Paradigm for the Primitive Relay Channel" *Algorithms* 12, no. 10: 218.
https://doi.org/10.3390/a12100218