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AdvDCSM System Tool and POLAR CODING

NOTE: This section is Under-Edit if necessary: Construction began on August 10, 2025 and was finished on .

POLAR BINARY CODES & SUCCESSIVE CANCELLATION DECODING: BPSK Signaling over a Coherent Memoryless Channel (MLC) 

by Darrell A. Nolta
August 10, 2025

The AdvDCSMT1DCSS (T1) Professional (T1 Version 2) 5G NR LDPCC PC Revision system tool has been used to create a set of Polar Codes (N, K)//Encoders and associated Successive Cancellation Decoders and investigate the phenomenon of Channel Polarization that was discovered by Erdal Arikan as described in his 2009 published paper (titled: 'Channel Polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels') .

The Polar Code belongs to the class of Linear Block Codes known as the Kronecker Codes. Fundamental to this class of codes is the Kronecker Matrix which is a Full-Rank Matrix. This matrix can be used to create a Kronecker Vector Space. The Kronecker Codes are contained within the subspaces of the Kronecker Vector space. The class of Kronecker Codes can be partitioned into two subclasses: 1) Reed Muller (RM) Codes and 2) Polar Codes. Key to the formation of the Polar Codes is the construction of the Kronecker Matrix. This construction is based on the use of multifold Kronecker products of a G2, a 2x2 binary matrix [[1,0],[1,1]].

A Polar Code is an example of a code with large N (CodeWord Length or Code Blocklength) that attempts to satisfy the C.E. Shannon's Noisy Channel Coding Theorem (i.e., the probability of decoding error approaches zero exponentially as the code blocklength is increased if the rate of information transmission is below the channel capacity).

NOTE: Shannon's Noisy Channel Coding Theorem does not provide a deterministic construction method.

The Communications (Comm) System Model used in this study consists of an Equiprobable IID Source for the K Information Bits, 5G NR Polar Codes (N, K) (Code Rate = 0.5; CodeWord Blocklength N = 8, 16, 32, 64, 128) Encoder without or with Universal Partial Order (UPO) (Universal Reliability Sequence) Use, BPSK Modulator & Demodulator for BPSK Signaling over a Coherent Memoryless Channel with Additive White Gaussian Noise (AWGN) and Successive Cancellation (SC) Decoder.

The Polar Code Encoder model specified by the 5G NR standard (3GPP set of TS 38.212 Version 16.2.0 Release 16 Standard) and used by T1 V2 can be described as the Alternative or Input G2 Kernels (IG2K) Polar Code Encoder model. This model is not the Arikan Polar Code Encoder Model.

The matching Polar Code Decoder is defined as the Alternative or Output G2 Kernels (OG2K) Polar Code Decoder model.

Note that T1 V2 supports the Arikan Polar Code Encoder Model and the Arikan Polar Code Decoder Model. Also, T1 V2 support Successive Cancellation List (SCL) Decoding and CRC Aided SCL Decoding.

One figure-of-merit of a Polar Coding System is: can it achieve the smallest Information Bit Error Rate (BER) (Pb) for a given transmit Information Bit Signal-to-Noise Ratio (SNR) Eb/N0. In the case of Polar Codes, Channel Polarization dictates whether or not a Polar Coding System will achieve the channel capacity and zero BER. Note that Channel Polarization creates a set of Bit-Coordinate Channels where each one has zero capacity and infinite BER (Noisy) and another set of Bit-Coordinate Channels where each one has a capacity of one and zero BER (i.e., Noiseless Channel).

Thus, if Channel Polarization is achieved in a given Polar Coding System, one wants to send Information Bits selectively over the most reliable Bit-Coordinate Channels. And, one wants to assign the least reliable Bit-Coordinate Channels to Frozen Bits.

Thus, for this Channel Polarization study, we have two cases: 1) using Consecutive Non-UPO Bit-Coordinate Channel-to-Frozen Bit assignment; and 2) using the 5G NR UPO standard to specify the UPO Bit-Coordinate Channel-to-Frozen Bit assignment.

The simulated Bit Error Rate (BER) (Pb) performance of each Comm system will be obtained and compared to the BER performance of a UnCoded BPSK Signaling over a Coherent Memoryless Channel with AWGN for each Information Bit Signal-to-Noise Ratio (Eb/N0) via a set of simulated Pb vs Eb/N0 graphs.

The seven simulated BER vs. Eb/N0 graphs Figures 1 through 7 are show below. Each figure of Figures 1 - 5 clearly shows the effect of using the 5G NR UPO method of choosing the set of Bit-Coordinate Channels to transmit the Information Bits. In fact for each figure of Figures 1 - 5, the BER performance for the Non-UPO case is worst than the BER performance for Maximum Likelihood (ML) Demodulation of UnCoded BPSK Signaling. And the BER performance for the UPO case is better than the BER performance for Maximum Likelihood (ML) Demodulation of UnCoded BPSK Signaling.

Figure 6 present the BER curves for all five Polar CodeWord BlockLengths (N = 8, 16, 32, 64, 128) and Non-UPO Bit-Coordinate Channel-to-Frozen Bit use (consecutive). One can clearly see that BER performance is worst for each curve (its curve is above the UnCoded BPSK Signaling Curve). This means that each Polar Code has lost its Error Correcting function. This observation is consistent with the theory of Channel Polarization. But what is very interesting is that as N is increased, the BER performance for a curve gets worst.

Figure 7 present the BER curves for all five Polar CodeWord BlockLength (N = 8, 16, 32, 64, 128) and UPO Bit-Coordinate Channel-to-Frozen Bit use (Non-consecutive). One can clearly see that BER performance is better for each curve (its curve is below the UnCoded BPSK Signaling Curve). This means that each Polar Code has gained its Error Correcting function. This observation is consistent with the theory of Channel Polarization. And, as N is increased, BER performance gets better. These observations are consistent with the theory of Channel Polarization.

But what is most important, the channel polarization to some degree appears to exist for Polar Code blocklengths that can be partitioned into SubSmall (8, 16, 32) and Small to Moderate (64 & 128) sets.

T1 Professional (T1 V2) 5G NR LDPCC PC Revision now offers the 5G NR Polar Codes in addition to 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 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.

This T1 V2 revision supports Gallager, Array, RA, Protograph-Based, 5G NR LDPC and 5G NR Polar Channel Coding for Signaling over a Memoryless Channel.

In conclusion, the User via T1 V2 5G NR LDPCC PC 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.

The User via T1 V2 5G NR LDPCC PC Revision can get experience with the use of Polar Codes and associated SC/SCL Decoders to achieve Channel Polarization and apply it to complex Digital Communication Systems for Spacecraft and Mobile Communications and Digital Storage Systems Polar Coding applications.



Figure 1. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 8, K = 4, Non-UPO or UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN:

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

5G NR Code (N = 8, K = 4, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model ;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive);

UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Successive Cancellation (SC) Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.

Note: BER Curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bits Errors):

N = 8, K = 4 5G NR Code UPO: 9, 10 dB.



Figure 2. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 16, K = 8, Non-UPO or UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN:

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

5G NR Code (N = 16, K = 8, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive);

UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Successive Cancellation (SC) Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.

Note: BER Curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bits Errors):

N = 16, K = 8 5G NR Code UPO: 9, 10 dB.



Figure 3. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 32, K = 16, Non-UPO or UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN:

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

5G NR Code (N = 32, K = 16, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive)

UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Successive Cancellation (SC) Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.

Note: BER Curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bits Errors):

N = 32, K = 16 5G NR Code UPO: 8, 9, 10 dB.



Figure 4. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 64, K = 32, Non-UPO or UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN:

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

5G NR Code (N = 64, K = 32, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive);

UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Successive Cancellation (SC) Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.

Note: BER Curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bits Errors):

N = 64, K = 32 5G NR Code UPO: 6, 7, 8, 9, 10 dB.



Figure 5. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 128, K = 64, Non-UPO or UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN:

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

5G NR Code (N = 128, K = 64, Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive);

UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Successive Cancellation (SC) Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.

Note: BER Curve's Eb/N0 dB Values for Pb = 0.0 (Zero Info Bits Errors):

N = 128, K = 64 5G NR Code UPO: 5, 6, 7, 8, 9, 10 dB.



Figure 6. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 8, 16, 32, 64, & 128; Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN:

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

5G NR Code (N = 8, K = 4; N = 16, K = 8; N = 32, K = 16; N = 64, K = 32; N = 128, K = 64; Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

Non-UPO Bit-Coordinate Channel-to-Frozen Bit Assignment (Consecutive);

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Successive Cancellation (SC) Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.



Figure 7. Bit Error Probability for Successive Cancellation Algorithm of 5G NR Polar Coded (N = 8, 16, 32, 64, & 128; UPO Bit-Coordinate Channel-to-Frozen Bit Assignment) BPSK Signaling over a Coherent Memoryless Channel (MLC) with AWGN:

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

5G NR Code (N = 8, K = 4; N = 16, K = 8; N = 32, K = 16; N = 64, K = 32; N = 128, K = 64; Code Rate = 0.5) implemented by a T1 V2 Alternative Encoder Model;

UPO Bit-Coordinate Channel-to-Frozen Bit Assignment per the 5G NR standard;

Maximum Likelihood (ML) Demodulation for UnCoded BPSK Signaling;

Successive Cancellation (SC) Algorithm is implemented by the T1 V2's Alternative Decoder using UnQuantized Messages.