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LDPC Codes: Addition of 5G NR Codes Addendum

NOTE: This section is Under-Edit if necessary: Construction began on December 30, 2024 and was finished on December 30, 2024.

Low-Density Parity-Check (LDPC) Binary Codes: Addition of 5G NR Codes Addendum 

by Darrell A. Nolta
December 30, 2024

The AdvDCSMT1DCSS (T1) Professional (T1 Version 2) 5G NR LDPCC Revision system tool has been used to investigate the Bit Error Rate (BER) performance difference between the use of the Generator Matrix Based Encoding and Parity-Check Matrix Based Encoding for the Sum-Product Algorithm (SPA) Decoding of Fifth-Generation Near Radio (5G NR) Low-Density Parity-Check (LDPC) Coded Signaling over 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.

Specifically, qualitative and quantitative BER performance comparisons have been made between the simulated use of Generator (G) Matrix Based Encoding as measured against the simulated use of Parity-Check (H) Matrix Based Encoding for a 5G NR LDPC, Punctured 5G NR LDPC, and Rate Matching Punctured 5G NR LDPC example codes (Base Graph (BG) 1, Lifting Size (Z) = 8, Parent CodeWord Length N = 544). This Encoding method comparison is based on the use of a simulated DT FFT-Based DMT Modulation Parallel MultiCarrier/MultiChannel (PMC) and Flooding Schedule SPA Decoder using Offset Min-Sum (OMS) Check (Ck) Messages.

Consult the 'Low-Density Parity-Check (LDPC) Binary Codes: Addition of Fifth Generation Near Radio (5G NR) Codes' paper on this website for details about T1 V2 5G NR LDPCC Revision system tool and the modeling and simulation of 5G NR LDPC Codes.

It is difficult to make a general statement about the similarity of the stream of CodeWords generated by the Generator Matrix Based Encoding and by the Parity-Check Matrix Based Encoding. The CodeWord's Information (Info) Bits segment for the G Matrix Based Encoding and H Matrix Based Encoding methods do match. The CodeWord's Parity-Check Bits segment for the G Matrix Based Encoding and H Matrix Based Encoding methods does not match most likely except for the case of an All-Zero CodeWord.

As a reminder, there two types of 5G NR LDPC Code Puncture: 1) Type 1 [UnLifted or Lifted Base Graph 1 Column 1 corresponding Z Info Bits are Deleted]; and 2) Type 2 [UnLifted or Lifted Base Graph 1 Column 1 & 2 corresponding 2xZ Info Bits are Deleted].

Consider the UnQuantized Flooding Scheduling SPA Decoding Algorithm Bit Error Rate (BER) or Bit Error Probability Pb performance simulation results for 5G NR Generator Matrix Based & Parity-Check Matrix Based Encoding and Signaling over a Coherent DT FFT-Based Discrete Multitone (DMT) Modulation Parallel MultiCarrier/MultiChannel (MC) that were produced by T1 V2 that are displayed below in Figure 1 plot. Further consider Figure 2 plot that shows the simulated BER results for Punctured 5G NR Generator Matrix Based & Parity-Check Matrix Based Encoding and Signaling over a Coherent DT FFT-Based DMT Modulation Parallel MC. Also, consider Figure 3 plot that show the simulated BER results for Rate Matching Punctured 5G NR Generator Matrix Based & Parity-Check Matrix Based Encoding and Signaling over a Coherent DT FFT-Based DMT Modulation PMC. The Signaling over the DT FFT-Based DMT Modulation PMC is based on M-ary modulations [BPSK, Orthogonal BPSK (PI/2 BPSK), Gray Coded QPSK, Gray Coded 16QAM] where each modulation is assigned to a carrier of the MultiCarrier/MultiChannel.

For comparison purposes, simulated BER results for UnCoded M-ary Signaling over a DT FFT-Based DMT Modulation Multicarrier/MultiChannel is included in Figures 1 through 3.

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 Offset Min-Sum (Offset = 1) Check Messages) of 5G NR (BG 1, Z = 8, N = 544) LDPC Coded (Generator Based & Parity-Check Matrix Encoding Based) 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 2 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Offset Min-Sum (Offset = 1) Check Messages) of Punctured (Type 2) 5G NR (BG 1, Z = 8, Transmit N = 528) 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 3 displays BER versus Eb/N0 values for Flooding Sum-Product Algorithm Decoding (using Offset Min-Sum (Offset = 1) Check Messages) of RM Punctured (Type 2) 5G NR NR (BG 1, Z = 8, 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 2) 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 transmitted blocklengh (N). Ng represents the number of CodeBits per group. 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 1-3 that is 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 figure taken from the Figures 1-3: 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 for 5G NR LDPC Code Encoding method 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 G Matrix Based Encoding as compared to H Matrix Based Encoding for use of the 5G NR code, Punctured (Type 2) 5G NR code, or Rate Matching Punctured (Type 2) 5G NR code.

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) OMS Check Messages for the selected T1 V2 5G NR LDPC example codes (Base Graph 1, Z = 8, N = 544) (Non Rate Matching Non Puncture, Puncture (Type 2), & Rate Matching Puncture (Type 2).

This conclusion is based on the fact that each 5G NR code BER curve {(Eb/N0,Pb)} in Figure 1 through Figure 3 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 a DT FFT-Based DMT Modulation PMC.

Next, we have determined for each figure taken from Figures 1 - 3 the BER performance curves' Eb/N0 order of the 5G NR LDPC Code (5GNRC) & Encoding method that is based on the figure's code & Encoding 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 [5GNRC]: Generator Matrix Based Encoding, Parity-Check Matrix Based Encoding;

Figure 2 [Punctured (Type 2) 5GNRC]: Generator Matrix Based Encoding, Parity-Check Matrix Based Encoding; &

Figure 3 [Rate Matching Punctured (Type 2) 5GNRC]: Generator Matrix Based Encoding, Parity-Check Matrix Based Encoding.

Next, we have determined the BER performance Coding Loss for Encoding Method for each figure taken from Figures 1-3. They are:

Figure 1 (5GNRC): Coding Loss [G Matrix Encoding - H Matrix Encoding, Reference (Ref)] = -0.04 dB;

Figure 2 (Punctured 5GNRC): Coding Loss [G Matrix Encoding - H Matrix Encoding, Reference (Ref)] = -0.06 dB; &

Figure 3 (RM Punctured 5GNRC): Coding Loss [G Matrix Encoding - H Matrix Encoding, Reference (Ref)] = -0.16 dB;

Using the BER performance Code Loss for Encoding Method boundary value of 0.25 dB, all three Coding Losses are negative which means that the Generator Matrix Based Encoding Method is the more efficient Encoding Method for the 5G NR example code, Punctured (Type 2) 5G NR example code, or Rate Matching Punctured (Type 2) 5G NR example code. But none of the Coding Losses for Encoding Method (absolute value) are greater than 0.25 dB.

So, in conclusion, it appears that for our example 5G NR LDPC Code (BG 1, Z = 8, N = 544), Punctured (Type 2) (Transmit N = 528) 5G NR LDPC Code or Rate Matching Punctured (Type 2) (Transmit N = 520) 5G NR LDPC code, the Generator Matrix Based and Parity-Check Matrix Based Encoding Methods are equivalent in regards to BER performance when Signaling over a DT FFT-Based DMT Modulation Parallel MultiCarrier/MultiChannel & Flooding SPA Decoding (Offset Min-Sum Check Messages) is used.

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 MultiCarrier/Multichannel. The Layered Sum-Product Algorithm (SPA) and the OMS Check Message scheme is supported by this T1 V2 revision 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 Comparison of Encoding Methods for Flooding 
Sum-Product Algorithm Decoding (using Offset Min-Sum Check Messages) of 5G NR 
(Base Graph 1, Lifting Size = 8, N = 544) 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;

Generator Matrix or Parity-Check Matrix Encoding;

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):

Generator Matrix Encoding (6.5, 7) & Parity-Check Encoding (6.5, 7).

FIGURE 2. Bit Error Probability Comparison of Encoding Methods for Flooding 
Sum-Product Algorithm Decoding (using Offset Min-Sum Check Messages) of Punctured 
5G NR (Base Graph 1, Lifting Size = 8, Transmit N = 528) 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 & Punctured 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 1 (Lifted BG 1 Column 1's 2xZ Info Bits Deleted): 
Transmit N = 528 & R = 0.333333;

Generator Matrix or Parity-Check Matrix Encoding;

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):

Generator Matrix Encoding (6, 6.5, 7) & Parity-Check Matrix Encoding (6.5,7).

FIGURE 3. Bit Error Probability Comparison of Encoding Methods for Flooding 
Sum-Product Algorithm Decoding (using Offset Min-Sum Check Messages) of Rate 
Matching (RM) Punctured 5G NR (Base Graph 1, Lifting Size = 8, 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 & RM Punctured 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;

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

Generator Matrix or Parity-Check Matrix Encoding;

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):

Generator Matrix Encoding (6, 6.5, 7) & Parity-Check Matrix Encoding (6.5,7).

      

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