# Minimum-Integer Computation Finite Alphabet Message Passing Decoder: From Theory to Decoder Implementations towards 1 Tb/s

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## Abstract

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## 1. Introduction

**x**, random vectors by bold sans-serif letters

**x**, realizations by serif letters x and vector-valued realizations by bold serif letters

**x**. Sets are denoted by calligraphic letters $\mathcal{X}$. The distribution ${p}_{\mathsf{x}}\left(x\right)$ of a random variable x is abbreviated as $p\left(x\right)$. $\mathsf{x}\to \mathsf{y}\to \mathsf{z}$ denotes a Markov chain, and $\mathbb{R}$, $\mathbb{Z}$, ${\mathbb{F}}_{2}$ denotes the real numbers, integers and Galois field 2, respectively.) are summarized as follows:

- We provide a novel criterion for the resolution of internal node operations to ensure that the MIC decoder can always replace the information maximizing VN mLUT exactly;
- we show that this MIC decoder has the same communication performance compared to an MI maximizing Min-mLUT decoder;
- we make an objective comparison between different FA-MP decoder implementations (Min-mLUT, Min-sLUT, MIC) in an advanced silicon technology and compare them with a state-of-the-art MS decoder for throughput towards 1 Tb/s;
- we show that our MIC decoder implementation outperforms state-of-the-art FP decoders in terms of routing complexity, area efficiency and energy efficiency and enables the processing of larger block sizes in state-of-the-art FP decoders since the routing complexity is largely reduced.

## 2. Preliminaries

#### 2.1. Transmission Model

**n**of variance ${\sigma}_{\mathsf{n}}^{2}$. A particular LDPC code is defined via a sparse parity check matrix $\mathit{H}\in {\mathbb{F}}_{2}^{M\times N}$. The Tanner graph [20] of an LDPC code is a visual representation of its parity check matrix

**H**and consists of a CN for each parity check equation ${\chi}_{m}$ with $m=1,...,M$ and a VN for each codebit ${c}_{n}$ with $n=1,...,N$. An edge connects VN n and CN m if and only if ${\mathit{H}}_{m,n}=1$. The degree of a node is determined by the number of connected edges. Furthermore, the fraction of edges that is connected to a node of a specific degree is characterized by the edge-degree distributions

#### 2.2. Iterative Decoding via Belief-Propagation (BP)

#### 2.3. Information Bottleneck Based Quantizer Design

## 3. LUT Decoder Design

#### 3.1. Check Node LUT Design

#### 3.2. Variable Node LUT Design

#### 3.3. Sequential LUT Design

## 4. Minimum-Integer Computation Decoder Design

- (i)
- mappings ${\varphi}_{V}$ and $\varphi $ of the ${n}_{\mathrm{E}}$-bit CN-to-VN messages ${a}_{m}$ and ${n}_{\mathrm{Q}}$-bit channel information z into node internal ${n}_{\mathrm{R}}$-bit signed integers with ${n}_{\mathrm{R}}\ge {n}_{\mathrm{E}}$ and ${n}_{\mathrm{R}}\ge {n}_{\mathrm{Q}}$, respectively;
- (ii)
- execution of integer additions for ${n}_{\mathrm{R}}$-bit signed integers;
- (iii)
- threshold quantization ${Q}_{V}$ to ${n}_{\mathrm{E}}$ bits determining the VN-to-CN message t.

**Figure 6.**VN update for computational domain framework [14,16]. The ${n}_{\mathrm{Q}}$-bit channel information $z\in \mathcal{Z}$ and the ${n}_{\mathrm{E}}$-bit CN-to-VN messages ${a}_{1},...,{a}_{{d}_{V}-1}\in \mathcal{A}$ are transformed to ${n}_{\mathrm{R}}$-bit signed integers. This transformation generally increases the required bit resolution for the representation, i.e., ${n}_{\mathrm{R}}\ge {n}_{\mathrm{Q}}$ and ${n}_{\mathrm{R}}\ge {n}_{\mathrm{E}}$. The internal signed integers are summed and quantized back into a ${n}_{\mathrm{E}}$-bit VN-to-CN message $t\in \mathcal{T}$.

#### 4.1. Equivalent LLR Quantizer

#### 4.2. Computations over Integers

#### Illustrative Example for MIC Calculations

#### 4.3. FER Results

## 5. Finite Alphabet Message Passing (FA-MP) Decoder Implementation

- NMS decoder with extrinsic message scaling factor of 0.75;
- Two LUT-based decoders: in these decoders, we implemented the VN operation by LUTs and the CN operations by a minimum search on the quantized messages. The latter corresponds to the CN Processor implementation of [7]. The LUTs are implemented either as a single LUT (mLUT), or as a tree of two-input LUTs (sLUT);
- Our new MIC decoder in which the VN is replaced by the new update algorithms, presented in the previous section.

#### 5.1. FER Performance of Implemented FA-MP Decoders

#### 5.2. FD-SOI Implementation Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Transmission model for transmission of LDPC encoded messages over an AWGN channel with quantization prior to FEC decoding.

**Figure 2.**Illustrative example of a CN update on a Tanner graph. The CN ${\chi}_{1}$ generates the CN-to-VN message ${L}_{n\leftarrow 1}$ for the VN ${c}_{n}$ based on the VN-to-CN messages ${L}_{1\to 1}$ and ${L}_{2\to 1}$ from VN ${c}_{1}$ and ${c}_{2}$, respectively.

**Figure 4.**Illustrative example for generation of extrinsic information in case of LUT decoding using discrete messages. (

**a**) visualizes a CN that generates the CN-to-VN message a based on incoming VN-to-CN messages ${t}_{1},...,{t}_{{d}_{C}-1}$. In (

**b**), a VN generates the VN-to-CN message t based on incoming CN-to-VN messages ${a}_{1},...,{a}_{{d}_{V}-1}$.

**Figure 5.**Graphical representation of a discrete CN update using ${n}_{\mathrm{E}}=4$ bit input messages ${t}_{1}$ and ${t}_{2}$ and a color-coded output message $a\in \mathcal{A}=\{-4,...,-1,1,...,4\}$. Subfigure (

**a**) shows the MI maximizing update ${f}_{3}^{\mathrm{MI}}({t}_{1},{t}_{2})$ and subfigure (

**b**) the minimum update ${f}_{3}^{min}({t}_{1},{t}_{2})$. The difference ${f}_{3}^{\mathrm{MI}}({t}_{1},{t}_{2})-{f}_{3}^{min}({t}_{1},{t}_{2})$ in subfigure (

**c**) contains only a few non-zero elemets and can be interpreted as a correction LUT.

**Figure 7.**Visualization of the relationship between the result of the calculation in the computational domain and the assignment to mutual information maximizing mLUT mapping. Subfigure (

**a**) shows the addition the real valued LLRs of (20) on the x-axis and the mutual information maximizing mLUT assignment of (16) on the y-axis. In Subfigure (

**b**–

**d**), the values on the x-axis are replaced by the corresponding integer additions of (21) for different scaling factors $s\in \{1,3,10\}$.

**Figure 8.**FER performance of ${n}_{\mathrm{E}}={n}_{\mathrm{Q}}=3$ bit Min-mLUT and MIC decoders using different internal message resolutions ${n}_{\mathrm{R}}$ for VN update.

**Figure 12.**Layout of decoders in the same scale; each color indicates one iteration stage from dark red (first iteration) to dark blue (eighth iteration).

**Table 1.**Numeric values of integer based VN update ${g}_{10}^{\mathrm{MIC}}$ with scaling parameter $s=10$.

$\mathrm{Cluster}\mathrm{index}z,a,t$ | $\pm 4$ | $\pm 3$ | $\pm 2$ | $\pm 1$ |
---|---|---|---|---|

Channel LLR $L\left(z\right)$ | $\pm 5.07$ | $\pm 2.90$ | $\pm 1.56$ | $\pm 0.49$ |

Integer mapping $\varphi \left(z\right)$ | $\pm 51$ | $\pm 29$ | $\pm 16$ | $\pm 5$ |

Message LLR $L\left(a\right)$ | $\pm 3.46$ | $\pm 2.08$ | $\pm 1.02$ | $\pm 0.25$ |

Integer mapping ${\varphi}_{V}\left(a\right)$ | $\pm 35$ | $\pm 21$ | $\pm 10$ | $\pm 3$ |

Interval ${\mathcal{W}}_{t}$ | $\pm \left(\right)open="["\; close="]">121,42$ | $\pm \left(\right)open="["\; close="]">41,25$ | $\pm \left(\right)open="["\; close="]">23,12$ | $\pm \left(\right)open="["\; close="]">11,1$ |

**Table 2.**Post-layout results of FA-MP decoders with ${n}_{\mathrm{E}}={n}_{\mathrm{Q}}=3$, ${n}_{\mathrm{R}}=5$.

MIC | Min-mLUT | Min-sLUT | NMS | |
---|---|---|---|---|

${n}_{\mathrm{E}}$, ${n}_{\mathrm{Q}}$ | 3 | 3 | 3 | 4 |

${E}_{b}/{N}_{0}$ @ FER ${10}^{-4}$ [dB] | $4.20$ | $4.16$ | $4.35$ | $4.26$ |

Utilization [%] | 70 | 68 | 71 | 71 |

Frequency [MHz] | 775 | 662 | 670 | 595 |

Coded Throughput [Gb/s] | 633 | 540 | 547 | 486 |

Area [mm^{2}] | $2.73$ | $4.23$ | $2.86$ | $3.04$ |

Area Efficiency [Gb/s/mm^{2}] | $231.6$ | 128 | 190 | $159.7$ |

Latency [ns] | $33.5$ | $39.3$ | $35.8$ | $43.7$ |

Power [W] | $4.49$ | $5.07$ | $4.38$ | $4.39$ |

Energy Efficiency [pJ/bit] | $7.10$ | $9.4$ | $8.0$ | $9.0$ |

**Table 3.**Post-layout results of FA-MP decoders with ${n}_{\mathrm{E}}={n}_{\mathrm{Q}}=4$, ${n}_{\mathrm{R}}=5$.

MIC | Min-mLUT | Min-sLUT | NMS | |
---|---|---|---|---|

${n}_{\mathrm{E}}$, ${n}_{\mathrm{Q}}$ | 4 | 4 | 4 | 5 |

${E}_{b}/{N}_{0}$ @ FER ${10}^{-4}$ [dB] | $3.94$ | $3.87$ | $3.93$ | $4.01$ |

Utilization [%] | 69 | 49 | 66 | 69 |

Frequency [MHz] | 633 | 267 | 492 | 183 |

Coded Throughput [Gb/s] | 516 | 218 | 401 | 149 |

Area [mm^{2}] | $3.66$ | $40.51$ | $7.82$ | $3.99$ |

Area Efficiency [Gb/s/mm^{2}] | $141.1$ | $5.4$ | $51.3$ | $37.4$ |

Latency [ns] | $41.1$ | $97.2$ | $48.0$ | $142.0$ |

Power [W] | $5.61$ | $11.85$ | $8.68$ | $2.25$ |

Energy Efficiency [pJ/bit] | $10.9$ | $54.3$ | $21.6$ | $15.1$ |

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**MDPI and ACS Style**

Monsees, T.; Griebel, O.; Herrmann, M.; Wübben, D.; Dekorsy, A.; Wehn, N.
Minimum-Integer Computation Finite Alphabet Message Passing Decoder: From Theory to Decoder Implementations towards 1 Tb/s. *Entropy* **2022**, *24*, 1452.
https://doi.org/10.3390/e24101452

**AMA Style**

Monsees T, Griebel O, Herrmann M, Wübben D, Dekorsy A, Wehn N.
Minimum-Integer Computation Finite Alphabet Message Passing Decoder: From Theory to Decoder Implementations towards 1 Tb/s. *Entropy*. 2022; 24(10):1452.
https://doi.org/10.3390/e24101452

**Chicago/Turabian Style**

Monsees, Tobias, Oliver Griebel, Matthias Herrmann, Dirk Wübben, Armin Dekorsy, and Norbert Wehn.
2022. "Minimum-Integer Computation Finite Alphabet Message Passing Decoder: From Theory to Decoder Implementations towards 1 Tb/s" *Entropy* 24, no. 10: 1452.
https://doi.org/10.3390/e24101452