# High Speed Decoding for High-Rate and Short-Length Reed–Muller Code Using Auto-Decoder

^{*}

## Abstract

**:**

## 1. Introduction

## 2. RM Decoder Based on AD

#### 2.1. Auto-Decoder

#### 2.2. RM Decoding Model

#### 2.3. Hyperparameters

#### 2.4. Performance Evaluation

## 3. PAD

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Symbol | Description |
---|---|

n | length of codeword (bits) |

r, m | parameters of RM code ($0\le r\le m$) |

k | length of message (bits) |

N | number of nodes in FC layer |

$\mathbf{y}$ | one-hot encoding vector |

$\mathbf{m}$ | message vector |

$\mathbf{z}$ | output of FC layer |

S | number of validation sets |

${\rho}_{t}$, ${\rho}_{v,s}$ | SNRs for the training set and the s-th validation set |

Layer | Number of Nodes |
---|---|

input | n |

1st hidden | $2n$ |

2nd hidden | $4n$ |

3rd hidden | $2n$ |

output | n |

loss function | cross-entropy |

optimizer | Adam |

training data set | ${2}^{k}\times {10}^{5}$ |

epoch | ${10}^{2}$ |

batch size | ${10}^{4}$ |

Training SNR (${\mathit{\rho}}_{\mathit{t}}$) | 0 | 1 | 2 | 3 |

NVE | 0.972 | 0.945 | 0.982 | 1.138 |

Training SNR (${\rho}_{t}$) | 4 | 5 | 6 | 7 |

NVE | 0.981 | 1.320 | 1.529 | 2.489 |

RM Code | Method | Time (ms) |
---|---|---|

$\mathfrak{R}(1,4)$ | FHT | 0.6012 |

AD | 0.3327 | |

$\mathfrak{R}(2,4)$ | FHT | 46.625 |

AD | 0.3704 |

Layer | Number of Nodes | ||||
---|---|---|---|---|---|

1st | 2nd | 3rd | 4th | 5th | |

CAD | CAD | CAD | CAD | CAD | |

1st hidden | n | $2n$ | $3n$ | $4n$ | $5n$ |

2nd hidden | n | $4n$ | $9n$ | $16n$ | $25n$ |

3rd hidden | n | $2n$ | $3n$ | $4n$ | $5n$ |

output | n | n | n | n | n |

Method | Time (ms) | Parameters | ||
---|---|---|---|---|

$\mathfrak{R}(\mathbf{1},\mathbf{4})$ | $\mathfrak{R}(\mathbf{2},\mathbf{4})$ | $\mathfrak{R}(\mathbf{1},\mathbf{4})$ | $\mathfrak{R}(\mathbf{2},\mathbf{4})$ | |

FHT | 0.6012 | 46.625 | - | - |

PAD-1 | 0.3327 | 0.3704 | 5808 | 40,080 |

PAD-2 | 0.3472 | 0.3785 | 6896 | 41,168 |

PAD-3 | 0.3838 | 0.4037 | 22,512 | 56,784 |

PAD-4 | 0.4075 | 0.4367 | 57,728 | 92,000 |

PAD-5 | 0.4520 | 0.4854 | 124,864 | 159,136 |

PAD-6 | 0.4972 | 0.5241 | 239,312 | 273,584 |

PAD-7 | 0.5508 | 0.6225 | 419,536 | 388,032 |

PAD-8 | 0.6198 | 0.6352 | 687,072 | 502,480 |

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**MDPI and ACS Style**

Cho, H.W.; Song, Y.J.
High Speed Decoding for High-Rate and Short-Length Reed–Muller Code Using Auto-Decoder. *Appl. Sci.* **2022**, *12*, 9225.
https://doi.org/10.3390/app12189225

**AMA Style**

Cho HW, Song YJ.
High Speed Decoding for High-Rate and Short-Length Reed–Muller Code Using Auto-Decoder. *Applied Sciences*. 2022; 12(18):9225.
https://doi.org/10.3390/app12189225

**Chicago/Turabian Style**

Cho, Hyun Woo, and Young Joon Song.
2022. "High Speed Decoding for High-Rate and Short-Length Reed–Muller Code Using Auto-Decoder" *Applied Sciences* 12, no. 18: 9225.
https://doi.org/10.3390/app12189225