# Low Delay Inter-Packet Coding in Vehicular Networks

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## Abstract

**:**

## 1. Introduction

#### 1.1. Safety Testing of C-ITS

#### 1.2. Roadside Units and Quasi Real-Time Transfers

#### 1.3. Convolutional Codes for Inter-Packet Coding

#### 1.4. Contributions

- New low-complexity low-delay decoding algorithm for erasure correction by the Wyner–Ash code applied in V2R scenario.
- Erasure-correcting performance analysis for Wyner–Ash and Reed–Solomon convolutional codes.
- Comparative analysis of suggested codes and decoding algorithms for: (i) memoryless channels; (ii) channels with memory described by Gilbert–Elliott model; and (iii) real-life VANET provided by AstaZero facility.

#### 1.5. Organization of the Paper

## 2. Preliminaries

#### 2.1. Wireless Channels

#### 2.2. Performance Metric

#### 2.3. Convolutional Codes for Network Applications

## 3. Packet Recovering Codes

#### 3.1. Binary Wyner–Ash Codes

#### 3.1.1. Code Description and Distance Properties

#### 3.1.2. Encoding

#### 3.1.3. Decoding

**Example**

**1.**

**Step****1.**- Compute syndrome. The syndrome is equal to [0 0 1]. From Equation (8) follows$$\left(\begin{array}{ccc}0& 0& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right).$$The number of unknowns is larger than the rank of the system which is equal to 2, that is, a unique solution does not exists. The decoder outputs only the information part of the first erasure-free block [1 1 0 0], i.e., output bits at this step are [1 1 0].
**Step****2.**- Shift the window. Input now is ${\mathit{y}}_{\mathrm{W}}=\left(\begin{array}{ccccc}0000& 1100& 1\varphi 0\varphi & 001\varphi & 0001\end{array}\right)$.The syndrome is equal to [0 0 1]. From Equation (8) follows$$\left(\begin{array}{ccc}1& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right).$$The unique solution is $[{z}_{1},{z}_{2},{z}_{3}]$ = [1 1 0]. The decoder decision is [1 1 0 1] and the output is [1 1 0]. At the next step the decoder will recover block [0 0 1 0] and the output bits are [0 0 1].

Algorithm 1 BP-BEC. |

while there exist parity checks with only one erased symbol do |

Assign to the erased symbol the modulo-2 sum of all nonerased symbols participating in the same parity check. |

end while |

#### 3.2. Nonbinary Convolutional Codes

#### 3.2.1. Code Description and Error-Correcting Properties

**Theorem**

**1.**

**Proof.**

- 1.
- Any erasure pattern $\mathit{\nu}=({\nu}_{1},{\nu}_{2},\dots ,{\nu}_{N})$ such that ${\nu}_{1}\le 3$ and ${\nu}_{i}=0$, $i\ne 1$ for any N will be corrected.
- 2.
- Any erasure pattern $\mathit{\nu}=({\nu}_{1},{\nu}_{2},\dots ,{\nu}_{N})$ such that ${\nu}_{i}<3$, ${\sum}_{i}{\nu}_{i}=3$ will be corrected.

**Theorem**

**2.**

**Proof.**

#### 3.2.2. Encoding and Decoding for the RS-Convolutional Codes

## 4. Numerical Results

#### 4.1. Memoryless Channel (BEC)

#### 4.2. Channel with Memory (M-BEC)

#### 4.3. Probability of Message Successful Delivering for AstaZero Scenario

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Comparison of SWML decoding and SWBP decoding with RPC of the binary Wyner–Ash code of rate $R=3/4$.

**Figure 4.**Comparison of SWBP, SWBP-RPC and SWML decoding of the binary Wyner–Ash code of rate $R=3/4$.

**Figure 6.**Comparison of rate 3/4 block codes and the WA convolutional code for different decoding delays.

**Figure 8.**Comparison of the binary rate $R=3/4$ WA code with the rate $R=3/4$ RS-convolutional code for different correlation coefficients of the fading channel.

**Figure 11.**SDF for the production zone. Simulation results for the WA codes of different rates used over production zone.

**Figure 12.**SDF for the entire session. Simulation results for the WA codes of different rates used over the entire session.

m | R | Spectrum Coefficients ${\mathit{g}}_{3},\dots ,{\mathit{g}}_{12}$ |
---|---|---|

2 | 3/4 | 6, 23, 80, 290, 1050, 3804, 13782, 49929, 180888, 655334 |

3 | 7/8 | 28, 275, 2456, 22468, 205826, 1885187, 17266158, 158138208, 1448368114, 13265417898 |

4 | 15/16 | 120, 2644, 52456, 1066592, 21738992, 442834486, 9021091078, 183772934474, 3743704654772, 76264411563598 |

**Table 2.**Coefficients of series expansion of $f\left(D\right)$ for the RS convolutional code of rate $R=3/4$.

m | R | Series Expansion Coefficients ${\mathit{f}}_{4},\dots ,{\mathit{f}}_{11}$ |
---|---|---|

2 | 3/4 | 1, 32, 342, 2282, 8756, 9657, −102562, −773838 |

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**MDPI and ACS Style**

Bocharova, I.; Kudryashov, B.; Lyamin, N.; Frick, E.; Rabi, M.; Vinel, A. Low Delay Inter-Packet Coding in Vehicular Networks. *Future Internet* **2019**, *11*, 212.
https://doi.org/10.3390/fi11100212

**AMA Style**

Bocharova I, Kudryashov B, Lyamin N, Frick E, Rabi M, Vinel A. Low Delay Inter-Packet Coding in Vehicular Networks. *Future Internet*. 2019; 11(10):212.
https://doi.org/10.3390/fi11100212

**Chicago/Turabian Style**

Bocharova, Irina, Boris Kudryashov, Nikita Lyamin, Erik Frick, Maben Rabi, and Alexey Vinel. 2019. "Low Delay Inter-Packet Coding in Vehicular Networks" *Future Internet* 11, no. 10: 212.
https://doi.org/10.3390/fi11100212