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LDPC Codes: Addition of Fifth-Generation New Radio (5G NR) Codes

NOTE: This section is Under-Edit if necessary: Construction began on December 7, 2024 and was finished on December 10, 2024.

Low-Density Parity-Check (LDPC) Binary Codes: Addition of Fifth-Generation Near Radio (5G NR) Codes 

by Darrell A. Nolta
December 10, 2024

The AdvDCSMT1DCSS (T1) Professional (T1 Version 2 5G NR LDPCC Revision 1) system tool has been used to investigate the Bit Error Rate (BER) performance for Sum-Product Algorithm (SPA) Decoding of Fifth-Generation Near Radio (5G NR) Low-Density Parity-Check (LDPC) Coded Signaling over a Coherent Memoryless Channel with Additive White Gaussian Noise (AWGN). Also, the BER performance for SPA Decoding of 5G NR LDPC Coded Signaling over a Rayleigh Fading Memory Channel with AWGN and a Coherent Discrete-Time (DT) ) Fast Fourier Transform (FFT)-Based Discrete Multitone (DMT) Modulation Parallel MultiCarrier/MultiChannel (MC) [Orthogonal frequency-division Multiplexing (OFDM) FFT-Based] with AWGN was investigated, too.

Specifically, qualitative and quantitative BER performance comparisons have been made between the simulated use of a Punctured or Rate Matching Punctured 5G NR LDPC example code as measured against the simulated use of a 5G NR LDPC example code (Parent CodeWord Length N = 544) for each simulated channel case (Memoryless, Rayleigh Fading, & DT FFT-Based DMT Modulation Parallel MultiCarrier/MultiChannel) and for Flooding Schedule SPA Decoding using Theoretical or Offset Min-Sum Check (Ck) Messages. Also, BER performance comparisons have been made between the simulated use of a Rate Matching Punctured 5G NR example code as measured against the simulated use of a Rate Matching 5G NR example code for the same channel and SPA decoding examples.

Note that in this study, the 5G NR LDPC Coding & SPA Decoding systems involve the use of the Parity-Check Matrix (PCM) (H) Based Encoding to produce the stream of CodeWords for transmission of Coded Information across each simulated channel case.

Also, note that Sum-Product Algorithm is a Multiple Iteration Soft Input/Soft-Decision Output (SISO) LDPC Code Channel Decoding algorithm that can be designed to operate with floating point or finite precision message processing.

T1 Version 2 has been revised to support the following new features:

1) 5G NR LDPC Code (Lifted Base Graph 1 or Lifted Base Graph 2) Computer-Based Generation;

2) 5G NR LDPC Code (Lifted Base Graph 1 or Lifted Base Graph 2) Computer-Based BER Performance Simulation;

3) 5G NR LDPC CodeWord Generation via Generator Matrix (G) Based Encoding or Parity-Check Matrix (H) Based Encoding during a 5G NR LDPC Coded Signaling BER Performance Simulation; &.

4) Punctured, Rate Matching (RM), or RM Punctured 5G NR LDPC Code (Lifted Base Graph 1 or Lifted Base Graph 2) Computer-Based BER Performance Simulation.

For a 5G NR LDPC Coded Signaling Simulation, the User can use a T1 V2 Rate Matching (RM) procedure with or without Puncture for Generator-Based 5G NR Encoding or Parity-Check Matrix-Based 5G NR Encoding. The T1 V2 Rate Model is based on the REDUNDANCY VERSION (RV) 0 model. This RM procedure will allow the 5G NR Coded Signaling Code Rate to be increased as compared to the parent 5G NR Code Rate.

There two types of 5G NR LDPC Code Puncture: 1) Type 1 [Lifted Base Graph 1 (2) Column 1 Info Bits Deleted]; and Type 2 [Lifted Base Graph 1 (2) Column 1 & 2 Info Bits Deleted].

Note: The Maximum Lifting Size (Z) is 384. But it is important to realize that the actual maximum Lifting Size for a particular Windows PC & T1 V2 combination is dependent on the memory capacity and speed of the Windows PC that T1 V2 is running on.

One of the channel coding applications of the Quasi-Cyclic Protograph-Based (PG) Codes is found in the Fifth-Generation Near Radio (5G NR) communications systems as described in the Third Generation Partnership Project (3GPP) TS 38.212 V16.20 (2020) standard. Consult the 3GPP Technical Specification document for 5G NR LDPC Code [1].

It is suggested that one review this website's paper titled "Low-Density Parity-Check (LDPC) Binary Codes: Addition of Protograph-Based Codes and Enhanced SPA Decoder Features (Layered Scheduling, Offset Min-Sum Check Message Passing, & Message Quantization) to obtain a further understanding of Structured Codes such as Quasi-Cyclic PG codes.

Actually the 5G NR Structured LDPC Code type is a Quasi-Cycle Protograph Code and a Low-Density Generator Matrix (LDGM) Code Concatenation. The 5G NR Single Edge (SE) Base Graph (BG) 1 or 2 is a ZEROS and ONES containing MATRIX that defines the set of BG edges. A SE BG edge is represented by a '1' (Single Check Node & Bit Node Connection) & a Non-Edge is denoted by a '0' for a given Check Node & Bit Node pair]. The 3GPP 5G NR standard specifies BG 1 and BG 2 set of BG edges and the corresponding Parity-Check Matrices.

The subject of 5G NR LDPC Codes (Structured LDPC Codes) is quite complicated and requires careful study.

A number of references have been used to implement the topics of this paper: they include 5G NR Codes 2018 paper [2], 2019 paper [3], L. Wang 2021 thesis [4]; & 5G NR codes Rate Matching 2018 paper [5], 2008 paper [6].

The actual BER performance determination of a implemented 5G NR LDPC Coded Signaling and Decoding system per the Third Generation Partnership Project (3GPP) TS 38.212 Standard is quite complex given the fact that it is dependent on the other system components (required by the Standard) . Thus, to establish a baseline BER performance, the 5G NR LDPC Coded Signaling and Decoding system has to be simplified by removing these additional components required by the 5G NR Multiplexing and Channel Coding Standard.

This addition of the new feature of 5G NR LDPC Codes' computer-based modeling and simulation to T1 V2 allows the User to conduct computer experiments to study these codes' fundamental BER performance for a variety of modulation schemes, channel types, and SPA decoding schemes.

This investigation/experiment can easily be conducted by AdvDCSMT1DCSS (T1) Professional (T1 V2) because of the capabilities of T1V2 that now include the above described features.

The T1 V2 5G NR LDPC Binary Codes capabilities/configuration for User Specified LDPC Codes is as follows:

Fifth-Generation Near Radio (5G NR) Codes: CodeWord Length N = 26112 MAX, Base Graph (Hb) 1 Size 46 x 68; N = 19968 MAX, Base Graph (Hb) 2 Size 42 x 52.

NOTE: The T1 V2 implemented 5G NR code's Maximum N is dependent on the User's computer capacity to run that code in practical time for a given number of transmitted Information Bits.

Consider the UnQuantized Flooding or Scheduling SPA Decoding Algorithm Bit Error Rate (BER) or Bit Error Probability Pb performance simulation results for 5G NR Parity-Check Matrix Encoding and Signaling that were produced by T1 V2 that are displayed below in Figure 1 through 7 plots for BPSK Modulation over a Coherent Memoryless Channel. Further consider Figure 8, 9, and 10 plots that show the simulated BER results for BFSK modulation over a Rayleigh Fading Memory Channel. Also, consider Figure 11, 12, and 13 plots that show the simulated BER results for M-ary modulation [BPSK, Orthogonal BPSK (PI/2 BPSK), Gray Coded QPSK, Gray Coded 16QAM] over a Coherent Discrete-Time (DT) FFT-Based Discrete Multitone (DMT) Modulation Parallel MultiCarrier/MultiChannel. (MC).

For comparison purposes, simulated BER results for UnCoded BPSK Signaling over a Coherent Memoryless Channel is included in Figures 1 through 7; UnCoded BFSK Signaling over a Raleigh Fading Memory Channel is included in Figures 8 through 10; and UnCoded M-ary Signaling over a DT FFT-Based DMT Modulation Multicarrier/MultiChannel is included in Figures 11 through 13.

Maximum Likelihood (ML) Demodulation was used for all three UnCoded Signaling-Channel cases (BPSK & MLC; BFSK & Rayleigh Fading Memory Channel; M-ary [BPSK, Orthogonal BPSK (PI/2 BPSK), Gray Coded QPSK, Gray Coded 16-QAM] & DT FFT-Based DMT Modulation Parallel MultiCarrier/MultiChannel].

Each figure's BER plot displays a number of curves where a curve is constructed from the set of simulated Pb values that correspond to a set of Eb/N0 [Signal-to-Noise Ratio (SNR)] values. Thus, a BER curve is represented as {(Eb/N0, Pb)} for a number of T1 V2 computer-based simulations. Each 5G NR code curve displays the BER performance behavior of a 5G NR Coding and SPA Iterative Decoding system example, respectively, that was used for a T1 V2 computer-based simulation.

Figure 1 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical Check Messages) of 5G NR (Base Graph (BG) 1, Lifting Size (Z) = 8, N = 544) & Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN; and for 1,000,332 Equal Probable Independent and Identical Distributed (I.I.D.) Information (Info) Bits.

Figure 2 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical Check Messages) of Rate Matching (RM) 5G NR (BG 1, Z = 8, Transmit N = 536,) & RM Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent MLC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 3 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical Check Messages) of 5G NR (BG 1, Z = 8, N = 544), Punctured & RM Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent MLC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 4 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Offset Min-Sum (Offset = 1) Check Messages) of 5G NR (BG 1, Z = 8, N = 544,) & Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent MLC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 5 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Offset Min-Sum (Offset = 1), Check Messages) of RM 5G NR (BG 1, Z = 8, Transmit N = 536) & RM Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent MLC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 6 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Offset Min-Sum (Offset = 1) Check Messages) of 5G NR (BG 1, Z = 8, N = 544), Punctured , & RM Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent MLC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 7 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical or Offset Min-Sum (Offset = 1) Check Messages) of Punctured 5G NR (BG 1, Z = 8, Transmit N = 528) & RM Punctured 5G NR (Transmit N = 520) LDPC Coded BPSK Signaling over a Coherent MLC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

The MLC has Non Distorting, UnRestricted Bandwidth.

Figure 8 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical Check Messages) of 5G NR (BG 1, Z = 8, N = 544), Punctured 5G NR, & RM Punctured 5G NR LDPC Coded BFSK Signaling over a Rayleigh Fading Memory Channel with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 9 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Offset Min-Sum (Offset = 1) Check Messages) of 5G NR (BG 1, Z = 8, N = 544), Punctured 5G NR & RM Punctured 5G NR LDPC Coded BFSK Signaling over a Rayleigh Fading Memory Channel with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 10 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical or Offset Min-Sum (Offset = 1) Check Messages) of Punctured 5G NR (BG 1, Z = 8, Transmit N = 528) & RM Punctured 5G NR (Transmit N = 520) LDPC Coded BFSK Signaling over a Rayleigh Fading Memory Channel with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

The User-specified Rayleigh Fading normalized 'Energy Gain' (dB) (Baseband Fading Average (Avg) Received SNR to Transmitted SNR (Es/N0) Ratio dB is specified as -5.25 dB.

The Rayleigh Fading Memory Channel has Non Distorting, UnRestricted Bandwidth.

Figure 11 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical Check Messages) of 5G NR (BG 1, Z = 8, N = 544), Punctured 5G NR, & RM Punctured 5G NR LDPC Coded M-ary Signaling over a Coherent Discrete-Time (DT) Fast Fourier Transform (FFT)-Based Discrete Multitone (DMT) Modulation Parallel MultiCarrier/MultiChannel (MC) with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 12 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Offset Min-Sum (Offset = 1) Check Messages) of 5G NR (BG 1, Z = 8, N = 544), Punctured 5G NR & RM Punctured 5G NR LDPC Coded M-ary Signaling over a Coherent DT FFT-Based DMT Modulation Parallel MC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

Figure 13 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Theoretical or Offset Min-Sum (Offset = 1) Check Messages) of Punctured 5G NR (BG 1, Z = 8, Transmit N = 528) & RM Punctured 5G NR (Transmit N = 520) LDPC Coded M-ary Signaling over a Coherent DT FFT-Based DMT Modulation Parallel MC with AWGN; and for 1,000,332 Equal Probable I.I.D. Info Bits.

In summary, the possible 5G NR LDPC Codes used to transmit Encoded Info Bits through the chosen channel are: 
5G NR Code (Parent): 0 of 544 CodeBits deleted for a Transmitted Code Rate 
(R) = 0.323529;

Punctured (Type 1) 5G NR Code: 8 of 544 CodeBits deleted for a Transmitted Code 
R = 0.328358;

Punctured (Type 2) 5G NR Code: 16 of 544 CodeBits deleted for a Transmitted Code 
R = 0.333333;

Rate Matching 5G NR Code: 8 of 544 CodeBits deleted for a Transmitted Code 
R = 0.328358;

Rate matching Punctured (Type 1) 5G NR Code: 16 of 544 CodeBits deleted for a 
Transmitted Code R = 0.333333; &

Rate Matching Punctured (Type 2) 24 of 544 CodeBits deleted for a Transmitted Code 
R = 0.338462.
As a reminder, a Low-Density Parity-Check Coded Parallel MultiCarrier/MultiChannel (MC) is partitioned into G parallel subcarrier/subchannel groups where a subcarrier/subchannel group consists of K parallel subcarrier/subchannels. The set {G * Ng} represents the possible partitions of the LDPC Code's blocklengh (N). This approach is used for the LDPC Code & Signaling over a Parallel MC application because N can be very large and the process of Codebits to Channel Input Bits assignment can quickly become unmanageable. Note {li} is the Group's set of the Number of Channel Input Bits.

The number of Inverse Discrete Fourier Transform Samples used in each 5G NR LDPC Coded Signaling over a DT FFT-Based DMT Modulation Parallel MC simulation is 1024. The number of Inverse Discrete Fourier Transform Samples used in each UnCoded Signaling over a DT FFT-Based DMT Modulation Parallel MC simulation is 8.

Note that for this experiment, T1 V2 used the FFT-Based DMT Modulation PMC MultiCarrier Signal transmitted over a Single Channel model (DMT Modulation PMC Type 0). Since this DT FFT-Based DMT Modulation PMC has Non Distorting, UnRestricted Bandwidth, no Cyclic Prefix (CP) is used.

Next, let us define the qualitative BER performance comparison where we order the BER curves for each figure taken from the Figures 1-13 based on each curve's Eb/N0 value at a Pb value of 1x10-4. The curve with the smallest Eb/N0 value will be ranked first.

Also, let us define a measure that will be used in the quantitative BER performance comparison between the subject BER curve {(Eb/N0,2, Pb, 2)} and the reference BER curve {(Eb/N0,1 Pb, 1)} for a chosen a figure taken from the Figures 1-13: the 'Coding Loss'. For Pb,2 = Pb,1 = 1x10-4, Code Loss = Eb/N0, 2 - Eb/N0,1. BER Interpolation is used to determine the Eb/N0 values for the Pb value of 1x10-4.

Let us assign a BER performance Coding Loss boundary value of 0.25 dB. Now if the subject-reference Coding Loss exceeds this value, then significant BER deterioration has occurred due to the use of the Punctured, Rate Matching, or Rate Matching & Punctured 5G NR code as compared to the use 5G NR code or to the use of Rate Matching Puncture 5G NR code as compared to Rate Matching 5G NR code.

There are a number of important conclusions that can be drawn from the below displayed simulated SPA Iterative Decoding of 5G NR LDPC Coded Signaling BER results.

First, it appears that T1 V2 is correctly modeling and simulating 5G NR LDPC Coded BPSK Signaling over a Coherent AWGN Memoryless Channel with Flooding SPA Decoding using UnQuantized (UnQ) Theoretical (T) or OMS Check Messages for the selected T1 V2 5G NR LDPC example code (Base Graph 1, Z = 8, N = 544) (Non Rate Matching &Non Puncture, Puncture (Type 1 & Type 2), Rate Matching, and Rate Matching Puncture (Type 1 & Type 2).

This conclusion is based on the fact that each 5G NR code BER curve {(Eb/N0,Pb)} in Figure 1 through Figure 7 exhibit the classic simulated BER curve behavior as a result of the use of Coded Signaling: partition into the Erroneous region (the curve's Pb is greater or equal to the Pb of UnCoded BPSK Signaling over a MLC); the 'waterfall' (a sharp reduction of Pb) or gradual reduction of Pb region; and Non-Erroneous region (the curve's Pb is exponential smaller than the Pb of UnCoded BPSK Signaling over MLC.

Next, we have determined for each figure taken from Figures 1-6 the BER performance curves' Eb/N0 order of the 5G NR LDPC Code (5GNRC) & SPA decoding method that is based on the figure's code & Flooding SPA decoding method curve's Eb/N0 at a Pb of 1x10-4 value. This ranking is based on the smallest to largest Eb/N0 values for a particular figure.

They are: Figure 1 [Theoretical (T) Ck Message]: 5G NRC, Punctured (P) (Type 1) 5GNRC, P (Type 2) 5GNRC;

Figure 2 (T Ck Message): Rate Matching (RM) 5GNRC, RM P (Type 1) 5GNRC, RM P (Type 2) 5GNRC;

Figure 3 (T Ck Message): 5GNRC, P (Type 2) 5GNRC, RM P Type 2 5GNRC;

Figure 4 [Offset OMS (OMS) Ck Message]: 5GNRC, P (Type 2) 5GNRC, P (Type 1) 5GNRC;

Figure 5 (OMS Ck Message): RM Punctured (Type 1) 5GNRC, RM 5GNRC, RM P (Type 2) 5GNRC; &

Figure 6 (OMS Ck Message): 5GNRC, P (Type 2) 5GNRC, RM P (Type 2) 5GNRC.

Next, we have determined the BER performance Coding Loss for each figure taken from Figures 1-6. They are:

Figure 1 (T Ck Message): Coding Loss [Punctured (P) (Type 1) 5GNRC - 5GNRC, Reference (Ref)] = 0.04 dB;

Figure 1 (T Ck Message): Coding Loss (P (Type 2) 5GNRC - 5GNRC, Ref) = 0.07 dB;

Figure 2 (T Ck Message): Coding Loss [RM P (Type 1) 5GNRC - RM 5GNRC, Ref] = 0.12 dB;

Figure 2 (T Ck Message): Coding Loss [RM P (Type 2) 5GNRC - RM 5GNRC, Ref] = 0.29 dB;

Figure 3 (T Ck Message): Coding Loss [P (Type 2) 5GNRC - 5GNRC, Ref] = 0.07 dB;

Figure 3 (T Ck Message): Coding Loss [RM P (Type 2) 5GNRC - 5GNRC, Ref] = 0.29 dB;

Figure 4 (OMS Ck Message): Coding Loss [P (Type 1) 5GNRC - 5GNRC, Ref] = 0.13 dB;

Figure 4 (OMS Ck Message): Coding Loss [P (Type 2) 5GNRC - 5GNRC, Ref] = 0.05 dB;

Figure 5 (OMS Ck Message): Coding Loss [RM P (Type 1) 5GNRC - RM 5GNRC, Ref] = -0.12 dB;

Figure 5 (OMS Ck Message): Coding Loss [RM P (Type 2) 5GNRC - RM 5GNRC, Ref] = 0.09 dB;

Figure 6 (T Ck Message): Coding Loss [P (Type 2) 5GNRC - 5GNRC, Ref] = 0.05 dB; &

Figure 6 (T Ck Message): Coding Loss [RM P (Type 2) 5GNRC - 5GNRC, Ref] = 0.19 dB.

Using the BER performance Code Loss boundary value of 0.25 dB, we have only two examples where their Coding Loss value exceeds this boundary value (indicates significant BER deterioration due to the use of Rate Matching Puncture was used). They are:

Figure 2 (T Ck Message): Coding Loss [RM P (Type 2) 5GNRC - RM 5GNRC, Ref] = 0.29 dB; &

Figure 3 (T Ck Message): Coding Loss [RM P (Type 2) 5GNRC - 5GNRC, Ref] = 0.29 dB.

Second, it appears that T1 V2 is correctly modeling and simulating 5G NR LDPC Coded BFSK Signaling over a Rayleigh Fading Memory Channel (MemC) with Flooding SPA Decoding using UnQuantized (UnQ) Theoretical (T) or OMS Check Messages for the selected T1 V2 5G NR LDPC example code (Base Graph 1, Z = 8, N = 544) (Non Rate Matching Non Puncture, Puncture (Type 1 & Type 2), Rate Matching, and Rate Matching Puncture (Type 1 & Type 2).

This conclusion is based on the fact that each 5G NR code BER curve {(Eb/N0,Pb)} in Figure 8 through Figure 10 exhibit the classic simulated BER curve behavior as a result of the use of Coded Signaling: partition into the Erroneous region (the curve's Pb is greater or equal to the Pb of UnCoded BFSK Signaling over a Rayleigh Fading MemC); the 'waterfall' (a sharp reduction of Pb) or gradual reduction of Pb region; and Non-Erroneous region (the curve's Pb is exponential smaller than the Pb of UnCoded BFSK Signaling over Rayleigh Fading MemC.

Next, we have determined for each figure taken from Figures 8 & 9 the BER performance curves' Eb/N0 order of the 5G NR LDPC Code (5GNRC) & SPA decoding method that is based on the figure's code & Flooding SPA decoding method curve's Eb/N0 at a Pb of 1x10-4 value. This ranking is based on the smallest to largest Eb/N0 values for a particular figure.

They are: Figure 8 [Theoretical (T) Ck Message]: 5GNRC, Punctured (P) (Type 2) 5GNRC, RM P (Type 2) 5GNRC;

Figure 9 [Offset OMS (OMS) Ck Message]: 5GNRC, RM P (Type 2) 5GNRC, P (Type 2) 5GNRC;

Next, we have determined the BER performance Coding Loss for each figure taken from Figures 8 & 9. They are:

Figure 8 (T Ck Message): Coding Loss [Punctured (P) (Type 2) 5GNRC - 5GNRC, Reference (Ref)] = 0.34 dB;

Figure 8 (T Ck Message): Coding Loss (RM P (Type 2) 5GNRC - 5GNRC Ref) = 0.5 dB;

Figure 9 (OMS Ck Message): Coding Loss [P (Type 2) 5GNRC - 5GNRC, Ref] = 0.47 dB; &

Figure 9 (OMS Ck Message): Coding Loss [RM P (Type 2) 5GNRC - 5GNRC, Ref] = 0.36 dB.

Using the BER performance Code Loss boundary value of 0.25 dB, all of the above examples' Coding Loss values exceeds this boundary value (indicates significant BER deterioration due to the use of Puncture or Rate Matching Puncture was used).

Third, it appears that T1 V2 is correctly modeling and simulating 5G NR LDPC Coded M-ary Signaling over a DT FFT-Based DMT Modulation Parallel MultiCarrier/MultiChannel with Flooding SPA Decoding using UnQuantized (UnQ) Theoretical (T) or OMS Check Messages for the selected T1 V2 5G NR LDPC example code (Base Graph 1, Z = 8, N = 544) (Non Rate Matching Non Puncture, Puncture (Type 1 & Type 2), Rate Matching, and Rate Matching Puncture (Type 1 & Type 2).

This conclusion is based on the fact that each 5G NR code BER curve {(Eb/N0,Pb)} in Figure 11 through Figure 13 exhibit the classic simulated BER curve behavior as a result of the use of Coded Signaling: partition into the Erroneous region (the curve's Pb is greater or equal to the Pb of UnCoded M-ary Signaling over a DT FFT-Based DMT Modulation PMC); the 'waterfall' (a sharp reduction of Pb) or gradual reduction of Pb region; and Non-Erroneous region (the curve's Pb is exponential smaller than the Pb of UnCoded M-ary Signaling over DT FFT-Based DMT Modulation PMC.

Next, we have determined for each figure taken from Figures 11 & 12 the BER performance curves' Eb/N0 order of the 5G NR LDPC Code (5GNRC) & SPA decoding method that is based on the figure's code & Flooding SPA decoding method curve's Eb/N0 at a Pb of 1x10-4 value. This ranking is based on the smallest to largest Eb/N0 values for a particular figure. They are:

Figure 11 [Theoretical (T) Ck Message]: Punctured (P) (Type 2) 5GNRC, 5G NRC, RM P (Type 2) 5GNRC; &

Figure 12 [Offset OMS (OMS) Ck Message]: 5GNRC, P (Type 2) 5GNRC, RM P (Type 2) 5GNRC.

Next, we have determined the BER performance Coding Loss for each figure taken from Figures 11-13. They are:

Figure 11 (T Ck Message): Coding Loss [Punctured (P) (Type 2) 5GNRC - 5GNRC, Reference (Ref)] = -0.01 dB;

Figure 11 (T Ck Message): Coding Loss (RM P (Type 2) 5GNRC - 5GNRC, Ref) = 0.17 dB;

Figure 12 (OMS Ck Message): Coding Loss [P (Type 2) 5GNRC - 5GNRC, Ref] = 0.12 dB; &

Figure 12 (OMS Ck Message): Coding Loss [RM P (Type 2) 5GNRC - 5GNRC, Ref] = 0.26 dB.

Using the BER performance Code Loss boundary value of 0.25 dB, we have only one example where their Coding Loss exceeds this boundary value (indicates significant BER deterioration due to the use of Rate Matching Puncture was used).

Finally, let us examine the effect on BER performance of the use the Flooding SPA Offset Min Sum (Offset = 1) Check Messages as compared to the use the Flooding SPA Theoretical Check Messages for each channel type:.

Eb/N0 Ranking at Pb = 1x10-4 & Coding Loss for BPSK Signaling over a Coherent Memoryless Channel case:

Figure 7 (T & OMS Ck Message): P (Type 2) 5GNRC & T, RM P (Type 2) 5GNRC & T, P (Type 2) 5GNRC & OMS, RM P (Type 2) 5GNRC & OMS.

Figure 7 (T & OMS Ck Message) Code Loss [5GNRC & OMS - 5GNRC & T, Ref] = 0.34 dB;

Figure 7 (T & OMS Ck Message): Code Loss [P (Type 2) 5GNRC & OMS - P (Type 2) 5GNRC & T, Ref] = 0.32 dB; &

Figure 7 (T & OMS Ck Message): Code Loss [RM P (Type 2) 5GNRC & OMS - RM P (Type 2) 5GNRC & T, Ref] = 0.27 dB.

We see that the use of OMS Check Message as compared to the use of Theoretical Check Messages for Flooding SPA Decoding causes significant BER deterioration for 5G NR, Puncture (Type 2) 5G NR, or Rate Matching (Type 2) 5G NR Coded BPSK Signaling over a Coherent Memoryless Channel (MLC).

Eb/N0 Ranking at Pb = 1x10-4 & Coding Loss for BFSK Signaling over a Rayleigh Fading Memory Channel:

Figure 10 (T & OMS Ck Message): P (Type 2) 5GNRC & T, RM P (Type 2) 5GNRC & T, RM P (Type 2) 5GNRC & OMS, P (Type 2) 5GNRC & OMS;

Figure 10 (T & OMS Ck Message) Code Loss [5GNRC & OMS - 5GNRC & T, Ref] = 1.83 dB;

Figure 10 (T & OMS Ck Message): Code Loss [P (Type 2) 5GNRC & OMS - P (Type 2) 5GNRC & T, Ref] = 1.96 dB; &

Figure 10 (T & OMS Ck Message): Code Loss [RM P (Type 2) 5GNRC & OMS - RM P (Type 2) 5GNRC & T, Ref] = 1.69 dB.

We see that the use of OMS Check Message as compared to the use of Theoretical Check Messages for Flooding SPA Decoding causes significant BER deterioration for 5G NR, Puncture (Type 2) 5G NR, or Rate Matching (Type 2) 5G NR Coded BFSK Signaling over a Rayleigh Fading Memory Channel (MemC).

Eb/N0 Ranking at Pb = 1x10-4 & Coding Loss for M-ary Signaling over a DT FFT-Based DMT Modulation Parallel MultiCarrier/MultiChannel:

Figure 13 (T & OMS Ck Message): P (Type 2) 5GNRC & T, RM P (Type 2) 5GNRC & T, P (Type 2) 5GNRC & OMS, RM P (Type 2) 5GNRC & OMS;

Figure 13 (T & OMS Ck Message) Code Loss [5GNRC & OMS - 5GNRC & T, Ref] = 1.37 db;

Figure 13 (T & OMS Ck Message): Code Loss [P (Type 2) 5GNRC & OMS - P (Type 2) 5GNRC & T, Ref] = 1.49 dB; &

Figure 13 (T & OMS Ck Message): Code Loss [RM P (Type 2) 5GNRC & OMS - RM P (Type 2) 5GNRC & T, Ref] = 1.46 dB.

We see that the use of OMS Check Message as compared to the use of Theoretical Check Messages for Flooding SPA Decoding causes significant BER deterioration for 5G NR, Puncture (Type 2) 5G NR, or Rate Matching (Type 2) 5G NR Coded BFSK Signaling over a DT FFT-Based DMT Modulation Parallel MultiCarrier/MultiChannel (PMC).

From the above described BER Performance results for the use of 5G NR LDPC Coded (example code: BG 1, Lifting Size = 8, N = 544) Signaling over each one of the three possible channel models (Coherent MLC, Rayleigh Fading MemC, & DT FFT-based DMT Modulation PMC), we have observed that 5G NR Coding and SPA Decoding provide strong forward error correction. When Puncture or Rate Matching Puncture 5G NR Coding is used, the observed BER behavior is complex at the Non-Erroneous region of the Eb/N0 domain. Further investigation is required where the Number of Information Bits being encoded is increased to 10 Million or 100 Million, This complex BER behavior is dependent on the nature of the channel as illustrated by observed BER behavior for the 5G NR Coded Signal over a Rayleigh Fading MemC. This particular 5G NR Code-Rayleigh Fading MemC combination has the largest Coding Losses.

T1 Professional (T1 V2) 5G NR LDPCC Revision now offers the 5G NR LDPC along with the Gallager, Array, Repeat-Accumulate (RA), and Permutation and Quasi-Cyclic Protograph-Based) LDPC codes construction. This T1 V2 revision supports Gallager, Array, RA, Protograph-Based, and 5G NR LDPC Channel Coding for Signaling over a Memoryless, Memory, or Parallel Multichannel. The Layered Sum-Product Algorithm (SPA) and the OMS Check Message scheme is supported by this T1 V2 revision in addition to the Flooding SPA and the Theoretical Check Message scheme for 5G NR Decoding. And, this T1 V2 revision supports the Quantization of SPA Channel Decoder Messages for 5G NR Coded Signaling over a MLC.

In conclusion, the User via T1 V2 5G NR LDPCC Revision can get experience with the Generation of 5G NR, and Gallager, Array, Repeat-Accumulate, Protograph-based (Permutation and Quasi-Cyclic) LDPC codes and the Sum-Product Algorithm as applied to Iterative Decoding in simulated Digital Communication Systems for Spacecraft and Mobile Communications and Digital Storage Systems LDPC Coding applications.

FIGURE 1  Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical Check Messages) of 5G NR (Base Graph 1, Lifting Size = 8, N = 544) & 
Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent Memoryless Channel (MLC) 
with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BPSK Signaling over a Coherent Vector MLC, 
respectively;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Theoretical Check Message 
ImplementationType; UnQuantized Messages; & Maximum Number of Iterations per Block 
(Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (2.5, 2.75, 3.0, 3.5);
Punctured (Type 1) 5G NR Code (2.35, 2.50, 2.75, 3.0, 3.5); &
Punctured (Type 2) 5G NR Code (2.75, 3.0, 3.5).

FIGURE 2. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical Check Messages) of Rate Matching (RM) 5G NR (Base Graph 1, Lifting Size 
= 8, Transmit N = 536,) & RM Punctured 5G NR LDPC Coded BPSK Signaling over a 
Coherent Memoryless Channel (MLC) with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & RM 5G NR LDPC Coded BPSK Signaling over a Coherent Vector MLC, 
respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

RM 5G NR Code (Z CodeBits Deleted): Transmit N = 536, & R = 0.328358;

RM Punctured (Type 1) 5G NR Code (Lifted BG 1 Column 1 Info Bits & Z CodeBits 
Deleted):Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2
(Check Messages then Bit Messages Iteration Processing); Theoretical Check Message 
Implementation Type; UnQuantized Messages; & Maximum Number of Iterations 
per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

RM 5G NR Code (2.75, 3.0, 3.5); RM Punctured (Type 1) 5G NR Code (2.75, 3.0, 3.5); 
& RM Punctured (Type 2) 5G NR Code (3.0, 3.5).

FIGURE 3. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical Check Messages) of 5G NR (Base Graph 1, Lifting Size = 8, N = 544), 
Punctured & Rate Matching (RM) Punctured 5G NR LDPC Coded BPSK Signaling over 
a Coherent Memoryless Channel (MLC) with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BPSK Signaling over a Coherent Vector MLC, 
respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): Transmit 
N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Theoretical Check Message 
Implementation Type; UnQuantized Messages; & Maximum Number of Iterations per Block 
(Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (2.5, 2.75, 3.0, 3.5); 
Punctured (Type 2) 5G NR Code (2.75, 3.0, 3.5); &
RM Punctured (Type 2) 5G NR Code (3.0, 3.5).

FIGURE 4. Bit Error Probability for Flooding Sum-Product Algorithm Decoding 
(using Offset Min-Sum Check Messages) of 5G NR  (Base Graph 1, Lifting Size = 8, 
N = 544,) & Punctured 5G NR LDPC Coded BPSK Signaling over a Coherent Memoryless 
Channel (MLC) with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BPSK Signaling over a Coherent Vector MLC, 
respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 1) 5G NR Code (Lifted BG 1 Column 1 Info Bits Deleted): Transmit 
N = 536 & R = 0.328358;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): 
Transmit N = 528 & R = 0.333333;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Offset Min-Sum [OMS, 
(Offset = 1)] Check Message Implementation Type; UnQuantized Messages; & Maximum 
Number of Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (2.75, 3.0, 3.5); 
Punctured (Type 1) 5G NR Code (3.0, 3.5); &
Punctured (Type 2) 5G NR Code (3.5).

FIGURE 5. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Offset Min-Sum Check Messages) of Rate Matching (RM) 5G NR (Base Graph 1, 
Lifting Size = 8, Transmit N = 536) & RM Punctured 5G NR LDPC Coded BPSK Signaling 
over a Coherent Memoryless Channel (MLC) with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & RM 5G NR LDPC Coded  BPSK Signaling over a Coherent Vector MLC, 
respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

RM 5G NR Code (Z CodeBits Deleted): Transmit N = 536, Base Graph (BG) 1, 
Lifting Size (Z) = 8, L = 176 & R = 0.328358;

RM Punctured (Type 1) 5G NR Code (Lifted BG 1 Column 1 Info Bits & Z CodeBits 
Deleted): Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code: Lifted BG 1 Column 1 & 2 Info Bits & Z Codebits 
Deleted): Transmit N = 520 & R = 0.338462; 

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Offset Min-Sum [OMS, 
(Offset = 1)] Check Message Implementation Type; UnQuantized Messages; & Maximum 
Number of Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

RM 5G NR Code (3.0, 3.5); 
RM Punctured (Type 1) 5G NR Code (3.5); & 
RM Punctured (Type 2) 5G NR Code (3.0, 3.5).

FIGURE 6. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Offset Min-Sum Check Messages) of 5G NR (Base Graph 1, Lifting Size = 8, N = 544), 
Punctured, & Rate Matching (RM) Punctured 5G NR LDPC Coded BPSK Signaling over a 
Coherent Memoryless Channel (MLC) with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BPSK Signaling over a Coherent Vector MLC, 
respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): 
Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462; 

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); ); Offset Min-Sum [OMS, 
(Offset = 1)] Check Message Implementation Type; UnQuantized Messages; & Maximum 
Number of Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (2.75, 3.0, 3.5); 
Punctured (Type 2) 5G NR Code (3.5); &
RM Punctured (Type 2) 5G NR Code (3.0, 3.5).

FIGURE 7. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical or Offset Min-Sum Check Messages) of Punctured 5G NR (Base Graph 1, 
Lifting Size = 8, Transmit N = 528) & Rate Matching (RM) Punctured 5G NR (Transmit 
N = 520) LDPC Coded BPSK Signaling over a Coherent Memoryless Channel (MLC) 
with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BPSK Signaling over a Coherent Vector MLC, 
respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): 
Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Theoretical (T) or Offset 
Min-Sum [OMS (Offset = 1)] Check Message Implementation Type; UnQuantized Messages; 
& Maximum Number of Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

Punctured (Type 2) 5G NR Code T (2.75, 3.0, 3.5); 
RM Punctured (Type 2) 5G NR Code T (3.0, 3.5);
Punctured (Type 2) 5G NR Code OMS (3.5); & 
RM Punctured (Type 2) 5G NR Code OMS (3.5).

FIGURE 8. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical Check Messages) of 5G NR (Base Graph 1, Lifting Size = 8, N = 544), 
Punctured 5G NR, & Rate Matching (RM) Punctured 5G NR LDPC Coded BFSK Signaling 
over a Rayleigh Fading Memory Channel with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BFSK Signaling over a Vector Rayleigh Fading 
Memory Channel, respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): 
Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462; 

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Maximum Likelihood (ML) Demodulation for UnCoded BFSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Theoretical Check Message 
Implementation Type; UnQuantized Messages; & Maximum Number of Iterations per Block 
(Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (16.35, 16.40, 16.5, 17, 17.5, 18, 19, 20);
Punctured (Type 2) 5G NR Code (16.5, 17, 17.5, 18, 19, 20); & 
RM Punctured (Type 2) 5G NR Code (18, 19, 20).

FIGURE 9. Bit Error Probability for Flooding Sum-Product Algorithm Decoding 
(using Offset Min-Sum Check Messages) of 5G NR (Base Graph 1, Lifting Size = 8, 
N = 544), Punctured 5G NR & Rate Matching (RM) Punctured 5G NR LDPC Coded BFSK 
Signaling over a Rayleigh Fading Memory Channel with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BFSK Signaling over a Vector Rayleigh Fading 
Memory Channel, respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): 
Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462;

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Maximum Likelihood (ML) Demodulation for UnCoded BFSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Offset Min-Sum [Offset = 1] 
Check Message Implementation Type; UnQuantized Messages; & Maximum Number of 
Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (18.5, 19, 19.5, 20); 
Punctured (Type 2) 5G NR Code (19, 19.5, 20); & 
RM Punctured (Type 2) 5G NR Code (20).

FIGURE 10. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical or Offset Min-Sum Check Messages) of Punctured 5G NR (Base Graph 1, 
Lifting Size = 8, Transmit N = 528) & Rate Matching (RM) Punctured 5G NR (Transmit 
N = 520) LDPC Coded BFSK Signaling over a Rayleigh Fading Memory Channel with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded BFSK Signaling over a Vector Rayleigh Fading 
Memory Channel, respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): 
Transmit N = 528 
& R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462; 

Rayleigh Fading (Frequency-NonSelective): -5.25 dB Normalized Energy Gain;

Maximum Likelihood (ML) Demodulation for UnCoded BFSK Signaling;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Theoretical (T) or Offset 
Min-Sum [OMS (Offset = 1)] Check Message Implementation Type; UnQuantized Messages; 
& Maximum Number of Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

Punctured (Type 2) 5G NR Code T (16.5, 17, 17.5, 18, 19, 20); 
RM Punctured (Type 2) 5G NR Code T (18, 19, 20);
Punctured (Type 2) 5G NR Code OMS (19, 19.5, 20); & 
RM Punctured (Type 2) 5G NR Code OMS (20).

FIGURE 11. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical Check Messages) of 5G NR (Base Graph 1, Lifting Size = 8, N = 544), 
Punctured 5G NR, & Rate Matching (RM) Punctured 5G NR LDPC Coded M-ary Signaling 
over a Coherent Discrete-Time (DT) FFT-Based Discrete Multitone (DMT) Modulation 
Parallel MultiChannel with AWGN:

Equal probable I.I.D Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded M-ary Signaling over a Coherent DT FFT-Based DMT 
Modulation Parallel MultiCarrier/MultiChannel, respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544 & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted); 
Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462;

The Distinct 4-MC Group Signaling Schemes consist of {li} = {1, 1, 2, 4} <=>
 {BPSK, PI/2 BPSK, QPSK, 16-QAM};

For each simulated Pb value, Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K), 
for 1 through K = 4 Signaling Schemes;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Theoretical Check 
Message Implementation Type; UnQuantized Messages; & Maximum Number of 
Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (5, 5.5, 6, 6.5, 7);
Punctured (Type 2) 5G NR Code (4.25, 4.5, 5, 5.5, 6, 6.5, 7); &
RM Punctured (Type 2) 5G NR Code (4.5, 5, 5.5, 6, 6.5, 7).

FIGURE 12. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Offset Min-Sum Check Messages) of 5G NR (Base Graph 1, Lifting Size = 8, N = 544), 
Punctured 5G NR & Rate Matching (RM) Punctured 5G NR LDPC Coded M-ary Signaling 
over a Coherent Discrete-Time (DT) FFT-Based Discrete Multitone (DMT) Modulation 
Parallel MultiChannel with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded M-ary Signaling over a Coherent DT FFT-Based DMT 
Modulation Parallel MultiCarrier/MultiChannel, respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted): 
Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): 
Transmit N = 520 & R = 0.338462; 

The Distinct 4-MC Group Signaling Schemes consist of {li} = {1, 1, 2, 4} <=> 
 {BPSK, PI/2 BPSK, QPSK, 16-QAM};

For each simulated Pb value, Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K), 
for 1 through K = 4 Signaling Schemes;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; 
Model 2 (Check Messages then Bit Messages Iteration Processing); Offset 
Min-Sum [Offset = 1] Check Message Implementation Type; UnQuantized Messages; 
& Maximum Number of Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

5G NR Code (6.5, 7); 
Punctured (Type 2) 5G NR Code (6.5, 7); & 
RM Punctured (Type 2) 5G NR Code (6.5, 7).

FIGURE 13. Bit Error Probability for Flooding Sum-Product Algorithm Decoding (using 
Theoretical or Offset Min-Sum Check Messages) of Punctured 5G NR (Base Graph 1, 
Lifting Size = 8, Transmit N = 528) & Rate Matching (RM) Punctured 5G NR 
(Transmit N = 520) LDPC Coded M-ary Signaling over a Coherent Discrete-Time (DT) 
FFT-Based Discrete Multitone (DMT) Modulation Parallel MultiChannel with AWGN:

Equal probable I.I.D. Source for 10 Million and 1,000,332 Information (Info) Bits 
for UnCoded & 5G NR LDPC Coded M-ary Signaling over a Coherent DT FFT-Based DMT 
Modulation Parallel MultiCarrier/MultiChannel, respectively;

5G NR Code (Parent): Base Graph (BG) 1, Lifting Size (Z) = 8, L = 176, N = 544, & 
Rate (R) = 0.323529;

Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits Deleted); 
Transmit N = 528 & R = 0.333333;

RM Punctured (Type 2) 5G NR Code (Lifted BG 1 Column 1 & 2 Info Bits & Z CodeBits 
Deleted): Transmit N = 520 & R = 0.338462;

The Distinct 4-MC Group Signaling Schemes consist of {li} = {1, 1, 2, 4} <=> 
 {BPSK, PI/2 BPSK, QPSK, 16-QAM};

For each simulated Pb value, Eb/N0 = Eb/N0(1) = Eb/N0(2) = … = Eb/N0(K), 
for 1 through K = 4 Signaling Schemes;

Sum-Product Algorithm (SPA) Iterative Decoder using Flooding Scheduling; Model 2 
(Check Messages then Bit Messages Iteration Processing); Theoretical (T) or Offset 
Min-Sum [OMS (Offset = 1)] Check Message Implementation Type; UnQuantized 
Messages; & Maximum Number of Iterations per Block (Imax) = 50.

Note: BER curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bit):

Punctured (Type 2) 5G NR Code T (4.25, 4.5, 5, 5.5, 6, 6.5, 7); 
RM Punctured (Type 2) 5G NR Code T (4.5, 5, 5.5, 6, 6.5, 7); 
Punctured (Type 2) 5G NR Code OMS (6.5, 7); &
RM Punctured (Type 2) 5G NR Code OMS (6.5, 7).
References:

[1] 3GPP (3rd Generation Partnership Project): Technical Specification 38.212 Version 16.20, 5G NR, Multiplexing and Channel Coding, 2020 (Online).

[2] Tom Richardson and Shrinivas Kudekar,"Design of Low-Density Parity Check Codes for 5G New Radio," IEEE Communications Magazine, pp. 28-34, March 2018.

[3] Jung Hyun Bae, Ahmed Abotabl, Hsien-Ping Lin, Kee-Bong Song and Jungwon Lee, "An Overview of channel coding for 5G NR cellular communication," SIP (2019), Vol 8, e17, 14 pages, Open Access Article.

[4] Lifang Wang, "Implementation of Low-Density Parity-Check codes for 5G NR shared channels," Degree Project in Computer Science & Engineering, School of Electrical Engineering & Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden, August 30, 2021.

[5] Fatemeh Hamidi-Sepehr, Ajit Nimbalker, and Gregory Ermolaev, “Analysis of 5G LDPC Codes Rate-Matching Design,” in 2018 IEEE 87th Vehicular Technology Conference (VTC Spring), 2018. doi:10.1109/VTCSpring.2018.8417496 pp. 1–5.

[6] Jung-Fu Cheng, Ajit Nimbalker, Yufei Blankenship, Brian Classon, T. Keith Blankenship, "Analysis of Circular Buffer Rate Matching for LTE Turbo Code," in 68th Vehicular Technology Conference, 2008, IEEE, pp. 1-5.

      

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