NOTE: This section is Under-Edit if necessary: Construction began on May 19, 2021 and was finished on May 20, 2020.

__Low-Density Parity-Check (LDPC) Binary Codes:
Irregular Gallager Coded Signaling over a Parallel MultiChannel and Iterative Message-Passing Channel Decoding__

by Darrell A. Nolta
May 20, 2021 with May 28, 2021 & November 1, 2021 Update

The

**AdvDCSMT1DCSS (T1) Professional (T1 Version 2) **system tool has been used to investigate the Bit Error Rate (BER) performance for Irregular Gallager Coded Signaling over a Discrete-Time Waveform Parallel MultiChannel (PMC) with Additive White Gaussian Noise. As a reminder, note that Regular and Irregular Gallager Codes are Low-Density Parity-Check (LDPC) Codes.

This investigation/experiment can easily be conducted by

**AdvDCSMT1DCSS (T1) Professional (T1 V2)** because of the capabilities of

**T1 V2** that are discussed in the paper 'Low-Density Parity-Check (LDPC) Binary Codes: Signaling over a Parallel MultiChannel and Iterative Message-Passing Channel Decoding.' This paper is found on this website.

An additional feature (function) has been added to

**T1 V2** which allows for the determination of a LDPC Code's Tanner Graph's Check Nodes' Degree (Number of Edges) Distribution and Bit Nodes' Degree Distribution. This feature counts the number of ones ('1') in each row (column) of the Code's Parity Check Matrix H for the Check (Bit) Nodes' Degree Distribution. This function is performed at the time of creation of the LDPC Code or is accessed via

**T1 V2** Design Menu.

The purpose of this investigation/experiment is to determine if an Irregular Gallager LDPC code's simulated BER performance can exceed the simulated BER performance of a Regular Gallager LDPC code (RGC). These Gallager LDPC codes belongs to a set (family) of codes that consist of one Regular Gallager Code (j = 3, k = 4, n = 504) that serves as a parent and a subset of Irregular Gallager Codes (IRRGC) that are derived from its parent RGC. Each one of these Gallager LDPC codes is

**T1 V2** Computer-generated. This determination is based on the comparison of BER performance for these codes where these codes are used in LDPC Coded Signaling over two basic Discrete-Time PMC channels: Additive White Gaussian Noise (AWGN) with AWGN and Discrete MultiTone (DMT) Modulation with AWGN.

**T1 V2** uses a model for the Discrete MultiTone (DMT) Modulation MultiCarrier/MultiChannel that is based on an Orthogonal frequency-division Multiplexing (OFDM) FFT-Based] with AWGN PMC model. There are two DMT models that

**T1 V2** has implemented:

LDPC Coded FFT-based DMT Discrete Time Waveform AWGN Modulation Channel Type: 0) MultiCarrier Signal transmitted over a Single Channel or 1) MultiCarrier Signal transmitted over a MultiChannel (MC) SubChannel.

Note: For the first time,

**T1 Professional** in this experiment uses

**T1 V2** features of Orthogonal Binary PSK (PI/2 BPSK) and Gray Coded Square 256-QAM for Low-Density Parity-Check Coded Signaling over a Parallel MultiChannel.

As a reminder, a Low-Density Parity-Check Coded Parallel MultiChannel (MC) is partitioned into G parallel subchannel groups where a subchannel group consists of K parallel subchannels. The set {G * N

_{g}} represents the possible partitions of the LDPC Code's blocklengh (N). This approach is used for the LDPC Code & Signaling over a PMC application because N can be very large and the process of Codebits to Channel Input Bits assignment can quickly become unmanageable. Note {l

_{i}} is the Group's set of the Number of Channel Input Bits.

This investigation used a

**T1 V2** capability of modeling and simulating Multiple Iteration Soft Input/Soft-Decision Output (SISO) LDPC Code Channel Decoding using the Sum-Product Algorithm (SPA). The SPA is a 'symbol-by-symbol' Maximum a Posteriori Probability (MAP) Belief Propagation Algorithm.

Consider the SPA Decoding Algorithm Bit Error Rate (BER) or Bit Error Probability P

_{b} performance simulation results for Regular and Irregular Gallager Coded (RGC) Generator-based Encoding and Signaling that were produced by

**T1 V2** that are displayed below in

**Figure 1** plot for the AWGN PMC. In addition, consider

**Figure 2 and 3** plots for the DMT Modulation PMC MultiCarrier Signal transmitted over a Single Channel and MultiCarrier Signal transmitted over a MultiChannel (MC) SubChannel, respectively.

For comparison purposes, the simulated BER results for UnCoded (UC) Distinct 8-MC Signaling over a Discrete-Time Waveform AWGN PMC (Non Distorting, UnRestricted Bandwidth,) are included in

**Figure 1**. The simulated BER results for UC Distinct 8-MC Signaling over Discrete-Time Waveform DMT Modulation PMC Type 0 and 1 (Non Distorting, UnRestricted Bandwidth,) are included in

**Figure 2 and 3**, respectively. These simulated BER results are based on the transmission of 10,000,032 Equal probable i.i.d. Info Bits. The Signaling Schemes consist of {l

_{i}} = {1,1,6,6,8,8,8,8} <=> {BPSK, PI/2 BPSK, 64-QAM, 64-QAM, 256-QAM, 256-QAM, 256-QAM, 256-QAM} for both PMCs. Note that Gray-coded Square signal constellation for 64-QAM and 256-QAM Signaling is used. A Maximum Likelihood Demodulation scheme is used.

Each figure's BER plot displays six curves where a curve is constructed from the set of simulated P

_{b} values that correspond to a set of E

_{b}/N

_{0} [Signal-to-Noise Ratio (SNR)] values. Thus, a BER curve is represented as {(E

_{b}/N

_{0}, P

_{b})} for a number of

**T1 V2** simulations. Each RGC (IRRGC) curve displays the BER performance behavior of a Regular (Irregular) Gallager Coding and SPA Iterative Decoding system example that was used for a

**T1 V2** simulation.

Since a PMC simulation consists of a number of Signaling Schemes (Distinct) the possible choices for the set of Signal Scheme's E

_{b}/N

_{0}^{(k)} values can become very large. To simplify this matter, for each P

_{b} simulation, all the Signaling Schemes' E

_{b}/N

_{0}^{(k)} values are specified so that they are all equal. Thus, a plot's E

_{b}/N

_{0} value is defined as

E

_{b}/N

_{0} = E

_{b}/N

_{0}^{(1)} = E

_{b}/N

_{0}^{(2)} = … = E

_{b}/N

_{0}^{(K)} , for 1 through K Signaling Schemes.

The five Gallager LDPC Coding Systems that were simulated to produce BER results for the AWGN PMC (as shown in

**Figure 1** ) are as follows:

1) [Number of Code Word Bits (N) = 504, column weight (j) = 3, row weight (k) = 4] Regular Gallager Code [Rate (R) = 0.253968], {G * N

_{g}} = {12 * 42}, 96 Subchannels, Distinct 8-MC Group (G), {l

_{i}} = {1,1,2,6,8,8,8,8} <=> {BPSK, PI/2 BPSK, QPSK, 64-QAM, 256-QAM, 256-QAM, 256-QAM, 256-QAM} for 1,000,064 Equal probable i.i.d. Information (Info) Bits;

2) [N = 507, j = 3, row weight vector: {k-1,k = 4,k+1}] Irregular Gallager Code (R = 0.258383), {G * N

_{g}} = {13 * 39}, 104 Subchannels, Distinct 8-MC G, {l

_{i}} = {1,1,1,4,8,8,8,8} <=> {BPSK, PI/2 BPSK, BPSK, 16-QAM, 256-QAM, 256-QAM, 256-QAM, 256-QAM} for 1,000,054 Equal probable i.i.d. Information (Info) Bits;

3) [N = 492, j = 3, row weight vector: {k-2,k-1,k = 4,k+1,k+2}] Irregular Gallager Code (R = 0.235772), {G * N

_{g}} = {12 * 41}, 96 Subchannels, Distinct 8-MC G, {l

_{i}} = {1,1,1,6,8,8,8,8} <=> {BPSK, PI/2 BPSK, BPSK, 64-QAM, 256-QAM, 256-QAM, 256-QAM, 256-QAM} for 1,000,036 Equal probable i.i.d. Information (Info) Bits;

4) [N = 506, j = 3, row weight vector: {k-1,k = 4,k+1}] Irregular Gallager Code (R = 0.256917), {G * N

_{g}} = {11 * 46}, 88 Subchannels, Distinct 8-MC G, {l

_{i}} = {1,1,6,6,8,8,8,8} <=> {BPSK, PI/2 BPSK, 64-QAM, 64-QAM, 256-QAM, 256-QAM, 256-QAM, 256-QAM} for 1,000,090 Equal probable i.i.d. Information (Info) Bits;

5) [N = 503, j = 3, row weight vector: {k-2,k-1,k = 4,k+1,k+2}] Irregular Gallager Code (R = 0.2552485), {G * N

_{g}} = {12 * 42}, 96 Subchannels, Distinct 8-MC G, {l

_{i}} = {1,1,2,6,8,8,8,8} <=> {BPSK, PI/2 BPSK, QPSK, 64-QAM, 256-QAM, 256-QAM, 256-QAM, 256-QAM} for 1,000,125 Equal probable i.i.d. Information (Info) Bits.

The same five Gallager LDPC Coding Systems as described above were simulated to produce BER results for the two DMT Modulation PMC models (as shown in

**Figure 2 and 3**) except Punctured BPSK was used instead of PI/2 BPSK. This Punctured BPSK modulation allowed a zero ('0') codebit to be transmitted and an one ('1') codebit to not be transmitted.

Note that Gray-coded Square signal constellation for QPSK, 16-QAM, 64-QAM and 256-QAM Signaling is used.

The following table shows the number of Input Channel Symbols transmitted per each CodeWord (Frame) and the total number of transmitted Frames for each Gallager code code for the AWGN PMC simulations.

Code N BPSK PI/2 BPSK QPSK 16-QAM 64-QAM 256-QAM No. of Frames
1) Regular 504 12 12 12 12 48 7813
2) Irregular 507 26 13 13 52 7634
3) Irregular 492 24 12 12 48 8621
4) Irregular 506 11 11 22 44 7693
5) Irregular 503 12 12 12 12 48 7875

For the simulations of DMT Modulation PMC models, PI/2 BPSK is replaced by a partial Punctured BPSK.

Note for the N = 503 Irregular Gallager Code, one of 96 Subchannels (Subchannel 0: BPSK) carries no Information. Thus, the Output Channel Symbols of the 95 subchannels is used by the SPA Decoding process. For the rest of the Gallager Codes, all of the Output Channel Symbols (corrupted Input Channel Symbols) are used by the SPA Decoding process.

**Figure 1** displays the BER versus E

_{b}/N

_{0} for the Maximum Number of Iterations per Block (Imax) of 50 for the five above described Gallager LDPC Coding Systems for Signaling over a Discrete-Time Waveform AWGN PMC (Non Distorting, UnRestricted Bandwidth) and using SPA Decoding.

**Figure 2** displays the BER versus E

_{b}/N

_{0} for the Maximum Number of Iterations per Block (Imax) of 50 for the five above described Gallager LDPC Coding Systems for Signaling over Discrete-Time Waveform DMT Modualtion PMC of LDPC Coded FFT-based DMT Discrete Time Waveform AWGN Modulation Channel Type 0 (MultiCarrier Signal transmitted over a Single Channel).

**Figure 3** displays the BER versus E

_{b}/N

_{0} for the Maximum Number of Iterations per Block (Imax) of 50 for the five above described Gallager LDPC Coding Systems for Signaling over Discrete-Time Waveform DMT Modulation PMC of LDPC Coded FFT-based DMT Discrete Time Waveform AWGN Modulation Channel Type 1 (MultiCarrier Signal transmitted over a MultiChannel (MC) SubChannel).

There are a number of important conclusions that can be drawn from the below displayed simulated Iterative LDPC Code SPA Decoding BER results in

**Figure 1, 2, and 3**. It appears that

__T1 is correctly modeling and simulating LDPC Coded Signaling over a Parallel MultiChannel/MultiCarrier Channel with AWGN with SPA Decoding for the selected Regular and Irregular Gallager Codes__.

**Figure 1** clearly shows that the reduction in BER as the SNR is increased for the Gallager Coded Signaling over an AWGN PMC based on Distinct 8-MC Group with SPA Decoding as compared to the UnCoded Signaling over a Distinct 8-MC AWGN PMC. Each Gallager Code BER curve exhibits the expected behavior as compared to the UC BER curve (i.e. a SNR region where the Gallager Code BER is greater than the UC BER, BER is equal to each other, and, then, a SNR region where the Gallager Code BER is less than the UC BER). Each Gallager Code BER curve exhibits a montonically decreasing curve with no fluctuations or error floor floor except for the N = 492 Irregular Gallager Code's BER curve where it begins to flatten out at SNR of 9.0 dB.

To judge the best Gallager Code's BER performance, the ranking will be based on Minimization of SNR at a BER of 10

^{-5}.

Rank Code N Rate {l_{i}} l^{eff} (Average value of {l_{i}})
Best Regular 504 0.253968 {1,1,2,6,8,8,8,8} 5.25
Irregular 503 0.255248 {1,1,2,6,8,8,8,8} 5.25
Irregular 506 0.256917 {1,1,6,6,8,8,8.8} 5.75
Irregular 507 0.258383 {1,1,1,4,8,8,8,8} 4.875
Worst Irregular 492 0.235772 {1,1,1,6,8,8,8,8} 5.125

**Figure 2** clearly shows that the reduction in BER as the SNR is increased for the Gallager Coded Signaling over a DMT Modulation PMC Type 0 based on Distinct 8-MC Group with SPA Decoding as compared to the UnCoded Signaling over a Distinct 8-MC DMT Modulation PMC. Each Gallager Code BER curve exhibits the expected behavior as compared to the UC BER curve (i.e. a SNR region where the Gallager Code BER is greater than the UC BER, BER is equal to each other, and, then, a SNR region where the Gallager Code BER is less than the UC BER). Each Gallager Code BER curve exhibits a montonically decreasing curve with no fluctuations or error floor except for the N = 492 and N = 503 Gallager Code BER curves where they begin to flatten out at SNR of 9.0 dB.

To judge the best Gallager Code's BER performance, the ranking will be based on Minimization of SNR at a BER of 10

^{-5}.

Rank Code N Rate {l_{i}} l^{eff} (Average value of {l_{i}})
Best Regular 504 0.253968 {1,1,2,6,8,8,8,8} 5.25
Irregular 507 0.258383 {1,1,1,4,8,8,8,8} 4.875
Irregular 506 0.256917 {1,1,6,6,8,8,8.8} 5.75
Irregular 503 0.255248 {1,1,2,6,8,8,8,8} 5.25
Worst Irregular 492 0.235772 {1,1,1,6,8,8,8,8} 5.125

**Figure 3** clearly shows that the reduction in BER as the SNR is increased for the Gallager Coded Signaling over a DMT Modulation PMC Type 1 based on Distinct 8-MC Group with SPA Decoding as compared to the UnCoded Signaling over a Distinct 8-MC DMT Modulation PMC. Each Gallager Code BER curve exhibits the expected behavior as compared to the UC BER curve (i.e. a SNR region where the Gallager Code BER is greater than the UC BER, BER is equal to each other, and, then, a SNR region where the Gallager Code BER is less than the UC BER). Each Gallager Code BER curve exhibits a montonically decreasing curve with no fluctuations or error floor.

To judge the best Gallager Code's BER performance, the ranking will be based on Minimization of SNR at a BER of 10

^{-5}.

Rank Code N Rate {l_{i}} l^{eff} (Average value of {l_{i}})
Best Irregular 507 0.258383 {1,1,1,4,8,8,8,8} 4.875
Regular 504 0.253968 {1,1,2,6,8,8,8,8} 5.25
Irregular 506 0.256917 {1,1,6,6,8,8,8.8} 5.75
Irregular 503 0.255248 {1,1,2,6,8,8,8,8} 5.25
Worst Irregular 492 0.235772 {1,1,1,6,8,8,8,8} 5.125

Next, we attempt to answer the question: can an Irregular Gallager LDPC code's BER performance exceed the BER performance of a Regular Gallager LDPC code (RGC) when signaling over an AWGN PMC, DMT Modulation PMC Type 0, or DMT Modulation PMC Type 1 and using Sum-Product Algorithm Decoding?

We observe that in only one PMC application, DMT Modulation PMC Type 1 (as shown in

**Figure 3**) that an Irregular Gallager LDPC Code's Simulated BER performance exceeds the simulated BER performance of its parent Regular Gallager LDPC Code.

One Million plus Info Bits for each Regular and Irregular Gallager Coded Signaling and SPA Decoding system

**T1V2** simulation was chosen to allow for reasonable simulation time to generate simulated BER results that can be use to make a 1st order comparison of BER performance between j = 3 Gallager Coded Signaling over a Parallel MultiChannel and SPA Decoding. Still, it must be realized that to obtain high confident BER results for long block code lengths, simulation time could take many hours or days (depending on the User's computer). Note that Dr. Gallager discusses this issue in his 1962 paper

**[1]** and 1963 book

**[2]**.

Let us look at the validity of our

**T1 V2** simulated results for AWGN PMC, DMT Modulation PMC Type 0, and DMT Modulation PMC Type 1 by comparing the degree of AWGN corruption of the Input Channel Symbols [via Gaussian Random Number (GRN) generation and addition] per each simulation:

I) AWGN PMC

Code N No. of GRNs GRN Mean GRN Variance Minimum GRN Value Maximum GRN Value
1) RGC 504 11,250,720 0.000188 0.499752 -3.591052 3.902831
2) IRRGC 507 11,115,104 0.000127 0.499974 -3.688250 3.777928
3) IRRGC 492 11,586,624 0.000221 0.499844 -3.597991 3.902831
4) IRRGC 506 10,154,760 0.000149 0.499844 -3.597991 3.902831
5) IRRGC 503 11,340,000 0.000180 0.499731 -3.688250 3.902831

Further, the generated minimum and maximum Gaussian RN values exceed 5 standard deviations.

II) DMT Modulation PMC Type 0

Code N No. of GRNs GRN Mean GRN Variance Minimum GRN Value Maximum GRN Value
1) RGC 504 4,000,256 -0.000128 0.499338 -3.478670 3.774967
2) IRRGC 507 3,908,608 -0.000137 0.499345 -3.478670 3.774967
3) IRRGC 492 4,413,952 -0.000125 0.499446 -3.478670 3.774967
4) IRRGC 506 3,938,816 -0.000198 0.499342 -3.478670 3.774967
5) IRRGC 503 4,032,000 -0.000203 0.499976 -3.688250 3.777928

Further, the generated minimum and maximum Gaussian RN values exceed 4.9 standard deviations.

III) DMT Modulation PMC Type 1

Code N No. of GRNs GRN Mean GRN Variance Minimum GRN Value Maximum GRN Value
1) RGC 504 3,000,192 -0.000525 0.499456 -3.478670 3.774967
2) IRRGC 507 3,175,744 -0.000113 0.499524 -3.441514 3.774967
3) IRRGC 492 3,310,464 0.000082 0.499331 -3.478670 3.774967
4) IRRGC 506 2,707,936 -0.000617 0.499881 -3.372414 3.774967
5) IRRGC 503 3,024,000 -0.000185 0.499905 -3.688250 3.774967

Further, the generated minimum and maximum Gaussian RN values exceed 4.7 standard deviations.

It appears from these Gaussian Random Number (RN) results for the five Gallager Codes and the three simulated PMC channel types, the Gaussian RN values generated by

**T1 V2** are an excellent model for baseband AWGN.

And, thus, the conclusions derived from

**Figure 1, 2, and 3** may be valid. It was found that in only one PMC set of simulations (DMT Modulation PMC Type 1) did an Irregular Gallager LDPC Code's Simulated BER performance exceed the simulated BER performance of its parent Regular Gallager LDPC Code (as shown in

**Figure 3**). To investigate these conclusions further, each Gallager Coding System simulation would be rerun with the initial number of Information (Info) Bits increased from 1 Million Info Bits to 10 Million Info Bits to obtain more reliable simulated BER results.

One must note that the analysis of BER performance of Gallager Codes (Parent Regular & Children Irregular) when used to signal over AWGN PMC, DMT Modulation PMC Type 0, and DMT Modulation PMC Type 1 channel is a very complex matter because of the diversity of 1-D & 2-D signaling schemes that can used. And for the family of Gallager Codes, the number of Frames used for each code will not agree with the rest of codes' number of Frames.

Plus, when we add in the complexity of the Iterative Message-Passing Channel Decoding (Sum-Product Algorithm) the analysis becomes unmanageable. One must realize that for each Frame (transmission of a CodeWord & decoding of a Corrupted CodeWord), the SPA Decoder begins by receiving the CodeBits' Reliability (Belief) data and then starts the Iterative process of Message-Passing (Belief Propagation) and SPA processing of the received beliefs.

But how does one do this analytically for the possible Gallager Coded Modulation choices and Parity-Check Matrix choices for possible PMC choices? A Gallager Code's parity-check matrix can be translated into the Tanner Graph that can be used to visualize the structure of this message-passing of beliefs along the edges of the Tanner Graph. It has been suggested that if one can compare the edge structure of each Gallager code, one might be able to compare qualitatively the BER performance of the codes

**[3]**]: there might be an improvement in BER performance and bit node degree profile differences. Also, there might be a relationship between check node degree profile and BER performance improvement.

Let us look at the Edge structure of the five Gallager Code's Tanner Graphs:

Code N No. of Bit Nodes with 3 Edges Check Node Edges (Degree)
1) RGC 504 504 No. of Check Nodes with Degree 4 = 378

2) IRRGC 507 507 No. of Check Nodes with Degree 3 = 120
No. of Check Nodes with Degree 4 = 129
No. of Check Nodes with Degree 5 = 129

3) IRRGC 492 492 No. of Check Nodes with Degree 2 = 93
No. of Check Nodes with Degree 3 = 69
No. of Check Nodes with Degree 4 = 81
No. of Check Nodes with Degree 5 = 51
No. of Check Nodes with Degree 6 = 84

4) IRRGC 506 506 No. of Check Nodes with Degree 3 = 117
No. of Check Nodes with Degree 4 = 138
No. of Check Nodes with Degree 5 = 123

5) IRRGC 503 503 No. of Check Nodes with Degree 2 = 75
No. of Check Nodes with Degree 3 = 69
No. of Check Nodes with Degree 4 = 87
No. of Check Nodes with Degree 5 = 78
No. of Check Nodes with Degree 6 = 69

From the above results, it is observed that the number of Edges for each Gallager Code's Bit Node is equal to three. And this is true for all five Gallager Codes. This approach does not help us here in the attempt to explain the Gallager Codes' BER Performance ranking. Also, the Check Node Degree distribution does not provide any insight into this matter either.

Thus, using

**T1 V2** capabilities for simulating Gallager Codes, Coded Modulated PMC Signaling, and SPA Decoding, one may be able to investigate the BER performance of these complex Coding structures.

**T1 Professional (T1 V2) now offers the LDPC Code (Gallager, Array, and Repeat-Accumulate) construction and LDPC Channel Coding and LDPC Channel Decoding (Iterative; based on the 'Symbol-by-Symbol' MAP Belief Propagation algorithm) for Parallel MultiChannel (PMC) features to the User.** The User can choose the simulated Signaling Channel as Additive White Gaussian Noise (AWGN) PMC with AWGN, Crosstalk (XTALK) PMC, or Discrete MultiTone (DMT) Modulation PMC with AWGN. Further, the user can choose the simulated PMC to possess a NonDistorting, UnRestricted Bandwidth or a Distorting, Restricted Bandwidth.

An important note to be recognized is that the DMT Modulation PMC is also known as Orthogonal frequency-division Multiplexing (OFDM).

**T1 V2's** OFDM implementation is FFT-Based.

Also, it is important to realize that New Radio (NR), a Fifth-Generation (5G) Telecommunications Technologies,
has been proposed to use LDPC Channel Codes

**[4]** & Cyclic Prefix (CP) OFDM (FFT-based)-based waveforms.

**T1 Professional** will provide the User the introductory opportunity to study the complexity of BER performance of the Low-Density Parity-Check Coded M-ary Signal over an OFDM (FFT-based) MultiCarrier/MultiChannel (with CP for Distorting, Restricted Bandwidth PMC or No CP for a NonDistorting, UnRestricted Bandwidth PMC) and Soft-Decision Decoding using the Sum-Product Algorithm.

In conclusion,

__the User via __**T1 V2** can get experience with the Generation of LDPC codes and the Sum-Product and Bit Flipping algorithms as applied to Iterative Decoding in simulated digital communication systems for Spacecraft and Mobile Communications and Digital Storage Systems LDPC Coding applications.__References:__**[1]** Robert G. Gallager, "Low-Density Parity-Check Codes,"

*IRE Transactions on Information Theory*, Vol. IT-8, pp. 21-28, January 1962.

**[2]** Robert G. Gallager,

*Low-Density Parity-Check Codes*, Number 21 of the M.I.T. Press Research Monographs, M.I.T. Press, Cambridge, Massachusetts, 1963.

**[3]** Daniel J. Costello, Jr., "An Introduction to Low-Density Parity Check Codes," https://my.ece.utah.edu/~rchen/courses/Costello-3.pdf, pp. 64-67, August 10, 2009.

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

**Figure 1.** Bit Error Probability for UnCoded, Regular, and Irregular Gallager
Coded Signaling over a Discrete-Time Waveform Additive White Gaussian Noise
(AWGN) Parallel MultiChannel (PMC) with AWGN:

Equal probable i.i.d. Source for 10,000,032, 1,000,064, 1,000,054, 1,000,036,
1,000,090 and 1,000,125 Information (Info) Bits for UnCoded; N = 504, k = 4
Regular Gallager Coded; N = 507, N = 492, N = 506, N = 503 Irregular Gallager
Coded Signaling respectively over a Discrete-Time Waveform (DTW) PMC Channel;

N = 504, j = 3, k = 4, L = 128, Rate = 0.253968 Regular Gallager Code (RGC)
(T1 V2 Computer-generated);

N = 507, j = 3, {k-1,k=4,k+1}, L = 131, Rate = 0.258383 Irregular Gallager Code
(IRRGC) (T1 V2 Computer-generated);
N = 492, j = 3, {k-2,k-1,k=4,k+1,k+2}, L = 116, Rate = 0.235772 IRRGC (T1 V2
Computer-generated);
N = 506, j = 3, {k-1,k=4,k+1}, L = 130, Rate = 0.256917 IRRGC (T1 V2
Computer-generated);
N = 503, j = 3, {k-2,k-1,k=4,k+1,k+2}, L = 127, Rate = 0.255248 IRRGC (T1 V2
Computer-generated);

For each simulated P_{b} value, E_{b}/N_{0} = E_{b}/N_{0}^{(1)} = E_{b}/N_{0}^{(2)} = … = E_{b}/N_{0}^{(K)}, for 1
through K Signaling Schemes;

Each UnCoded or Gallager Coded DTW AWGN PMC subchannel consists of half-cosine
orthonormal baseband shaping pulse, 8 symbols per symbol period, and half-cosine
matched filter demodulator front-end;

These DTW subchannels possess a NonDistorting, UnRestricted Bandwidth; &

**Sum-Product Algorithm Iterative Decoder** using Model 2 (Check Messages then
Bit Messages Iteration Processing), Theoretical SPA Implementation Type, and
Maximum Number of Iterations per Block (Imax) = 50.

**Figure 2.** Bit Error Probability for UnCoded, Regular, and Irregular Gallager
Coded Signaling over a Discrete-Time Discrete MultiTone (DMT) Modulation Parallel
MultiCarrier/MultiChannel (PMC) Type 0 with Additive White Gaussian Noise (AWGN):

Equal probable i.i.d. Source for 10,000,032, 1,000,064, 1,000,054, 1,000,036,
1,000,090 and 1,000,125 Information (Info) Bits for UnCoded; N = 504, k = 4
Regular Gallager Coded; N = 507; N = 492, N = 506, N = 503 Irregular Gallager
Coded Signaling respectively over a Discrete-Time (DT) DMT Modulation PMC Channel
Type 0;

N = 504, j = 3, k = 4, L = 128, Rate = 0.253968 Regular Gallager Code (RGC)
(T1 V2 Computer-generated);

N = 507, j = 3, {k-1,k=4,k+1}, L = 131, Rate = 0.258383 Irregular Gallager Code
(IRRGC) (T1 V2 Computer-generated);
N = 492, j = 3, {k-2,k-1,k=4,k+1,k+2}, L = 116, Rate = 0.235772 IRRGC (T1 V2
Computer-generated);
N = 506, j = 3, {k-1,k=4,k+1}, L = 130, Rate = 0.256917 IRRGC (T1 V2
Computer-generated);
N = 503, j = 3, {k-2,k-1,k=4,k+1,k+2}, L = 127, Rate = 0.255248 IRRGC (T1 V2
Computer-generated);

For each simulated P_{b} value, E_{b}/N_{0} = E_{b}/N_{0}^{(1)} = E_{b}/N_{0}^{(2)} = … = E_{b}/N_{0}^{(K)}, for 1
through K Signaling Schemes;

These DT DMT Modulation PMC Type 0 channels possess a NonDistorting, UnRestricted
Bandwidth; &

**Sum-Product Algorithm Iterative Decoder** using Model 2 (Check Messages then
Bit Messages Iteration Processing), Theoretical SPA Implementation Type, and
Maximum Number of Iterations per Block (Imax) = 50.

**Figure 3.** Bit Error Probability for UnCoded, Regular, and Irregular Gallager
Coded Signaling over a Discrete-Time Discrete MultiTone (DMT) Modulation Parallel
MultiCarrier/MultiChannel (PMC) Type 1 with Additive White Gaussian Noise (AWGN):

Equal probable i.i.d. Source for 10,000,032, 1,000,064, 1,000,054, 1,000,036,
1,000,090 and 1,000,125 Information (Info) Bits for UnCoded; N = 504, k = 4
Regular Gallager Coded; N = 507; N = 492, N = 506, N = 503 Irregular Gallager
Coded Signaling respectively over a Discrete-Time (DT) DMT Modulation PMC Channel
Type 1;

N = 504, j = 3, k = 4, L = 128, Rate = 0.253968 Regular Gallager Code (RGC)
(T1 V2 Computer-generated);

N = 507, j = 3, {k-1,k=4,k+1}, L = 131, Rate = 0.258383 Irregular Gallager Code
(IRRGC) (T1 V2 Computer-generated);
N = 492, j = 3, {k-2,k-1,k=4,k+1,k+2}, L = 116, Rate = 0.235772 IRRGC (T1 V2
Computer-generated);
N = 506, j = 3, {k-1,k=4,k+1}, L = 130, Rate = 0.256917 IRRGC (T1 V2
Computer-generated);
N = 503, j = 3, {k-2,k-1,k=4,k+1,k+2}, L = 127, Rate = 0.255248 IRRGC (T1 V2
Computer-generated);

For each simulated P_{b} value, E_{b}/N_{0} = E_{b}/N_{0}^{(1)} = E_{b}/N_{0}^{(2)} = … = E_{b}/N_{0}^{(K)}, for 1
through K Signaling Schemes;

These DT DMT Modulation PMC Type 1 channels possess a NonDistorting, UnRestricted
Bandwidth; &

**Sum-Product Algorithm Iterative Decoder** using Model 2 (Check Messages then
Bit Messages Iteration Processing), Theoretical SPA Implementation Type, and
Maximum Number of Iterations per Block (Imax) = 50.

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