Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Allerton Conference on Communication, Control, and Computing

On bit error rate performance of polar codes in finite regime

Ali Eslami, Hossein Pishro-Nik

Published Sep 1, 2010• 53 citations
Publisher

Abstract

Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve the capacity of symmetric binary-input memoryless channels. Here, we study the bit error rate performance of finite-length polar codes under Belief Propagation (BP) decoding. We analyze the stopping sets of polar codes and the size of the minimal stopping set, called “stopping distance”. Stopping sets, as they contribute to the decoding failure, play an important role in bit error rate and error floor performance of the code. We show that the stopping distance for binary polar codes, if carefully designed, grows as O(√N) where N is the code-length. We provide bit error rate (BER) simulations for polar codes over binary erasure and gaussian channels, showing no sign of error floor down to the BERs of 10-11. Our simulations asserts that while finite-length polar codes do not perform as good as LDPC codes in terms of bit error rate, they show superior error floor performance. Motivated by good error floor performance, we introduce a modified version of BP decoding employing a guessing algorithm to improve the BER performance of polar codes. Our simulations for this guessing algorithm show two orders of magnitude improvement over simple BP decoding for the binary erasure channel (BEC), and up to 0.3 dB improvement for the gaussian channel at BERs of 10-6.

DOI
10.1109/allerton.2010.5706906

Related papers

Enhanced Belief Propagation Polar Decoder for Finite Lengths

Shajeel Iqbal, GoangSeog Choi

In this paper, we discuss the belief propagation decoding algorithm for polar codes. The performance of Polar codes for shorter lengths is not satisfactory. Motivated by this, we propose a novel technique to improve its performance at short lengths. We showed that the probability of messages passed along the factor graph of polar codes, can be increased by multiplying the current message of nodes with their previous message. This is like a feedback path in which the present signal is updated by multiplying with its previous signal. Thus the experimental results show that performance of belief propagation polar decoder can be improved using this proposed technique. Simulation results in binary-input additive white Gaussian noise channel (BI-AWGNC) show that the proposed belief propagation polar decoder can provide significant gain of 2 dB over the original belief propagation polar decoder with code rate 0.5 and code length 128 at the bit error rate (BER) of $10^{-4}$ .

Journal of the Korea Society of Digital Industry and Information Management•Sep 30, 2015•0 citations

Assessing the effectiveness of polar coding with BP and BPL decoding at different Eb/No values in a BEC

Zahraa Abbas Hamza, Shereen Abdalkadum Shandal

Polar Codes are a new class of capacity-reaching codes that have low decoding complexity and excessive reliability. They are primarily based on the concept of channel polarization, which makes a few bits inside the codeword more reliable and a few much less reliable because the code duration increases. This examination comprehensively evaluates how effective Belief Propagation (BP) and Belief Propagation with List (BPL) Decoding algorithms are in the Binary Erasure Channel (BEC) setup. The fundamental purpose is to recognize how nicely these algorithms deal with errors and enhance the reliability of verbal exchange structures. Different code lengths and list sizes are examined to see how they affect the Block Error Rate (BLER). This paper considers 4 codes studied and carried out in the usage of MATLAB lengths: N=256, 512, 1024, and 2048, and varies the Energy per bit to Noise power (Eb/No) from 1 dB to a 3.5 dB with list sizes eight and sixteen. The experiments show how the BP and BPL algorithms perform in coping with erasure errors and ensuring reliable deciphering. The BLER figures provide an in-depth description and comparison of the BP and BPL interpreting algorithms. The analysis highlights the promising blunder-correction talents of each BP and BPL algorithm in the BEC context. As the code length increases, the capability to correct errors also improves, showing that these algorithms can scale effectively. The addition of list decoding in BPL similarly enhances blunder correction by offering multiple possible answers. These examine the effect of Eb/No stages on the performance of BP and BPL deciphering, displaying that better Eb/No stages enhance accuracy but also increase the erasure chance. This paper discusses the trade-offs between accuracy and complexity for BP and BPL deciphering, presenting insights into their performance traits and optimization.

AIP conference proceedings•Jan 1, 2024•0 citations

Frozen bits selection for polar codes based on simulation and BP decoding

Jingbo Liu, Jin Sha

In the construction of polar code, the selection of frozen bits affects the error-correcting performance significantly. Several calculation-based algorithms have been proposed for general binary-input discrete memoryless channels (B-DMCs) like the additive white Gaussian noise (AWGN) channel. In this paper, a method for frozen bits selection based on Monte Carlo simulation and belief propagation (BP) decoding is proposed. The information bits are selected out one by one incrementally. The numerical results show that the proposed algorithm can effectively improve the bit error rate (BER) and frame error rate (FER) performance compared with the conventional selection method, especially in high signal noise ratio (SNR) region. Moreover, the algorithm can be used to construct polar codes with any rate through a complete iteration.

IEICE Electronics Express•Jan 1, 2017•13 citations

Finite-length analysis of turbo decoding on the binary erasure channel

Eirik Rosnes, Øyvind Ytrehus

This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation (BP) decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of stopping sets. In the first part we describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or until the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum size turbo stopping set which is also a subset of the set of erased positions. In the second part we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results

IEEE International Symposium on Information Theory (ISIT)•Jan 1, 2005•4 citations

Upper Bounding the Performance of Arbitrary Finite LDPC Codes on Binary Erasure Channels

Chih-chun Wang, Sanjeev R. Kulkarni, H. Vincent Poor

Assuming iterative decoding for binary erasure channels (BECs), a novel tree-based technique for upper bounding the bit error rates (BERs) of arbitrary, finite low-density parity-check (LDPC) codes is provided and the resulting bound can be evaluated for all operating erasure probabilities, including both the waterfall and the error floor regions. This upper bound can also be viewed as a narrowing search of stopping sets, which is an approach different from the stopping set enumeration used for lower bounding the error floor. When combined with optimal leaf-finding modules, this upper bound is guaranteed to be tight in terms of the asymptotic order. The Boolean framework proposed herein further admits a composite search for even tighter results. For comparison, a refinement of the algorithm is capable of exhausting all stopping sets of size les 13 for irregular LDPC codes of length n ap 500, which requires (13500) ap 1.67 times 1025trials if a brute force approach is taken. These experiments indicate that this upper bound can be used both as an analytical tool and as a deterministic worst-performance (error floor) guarantee, the latter of which is crucial to optimizing LDPC codes for extremely low BER applications, e.g., optical/satellite communications

arXiv:cs/0605086•IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2006•8 citations

Belief Propagation With Permutated Graphs of Polar Codes

Liping Li, Liangliang Liu

The belief propagation (BP) decoding of polar codes provides better bit error rate (BER) and packet error rate (PER) performance than the successive cancelation (SC) decoding. It has been reported in the literature that the graph of BP decoding of polar codes can be permuted to provide even better decoding performance. In this paper, we theoretically prove that all permutations of the graph are equivalent in terms of the polar encoding. For BP decoding, it is shown that permutations of the layers of the graph also permute the bit channels. A principle which protects the weakest information bit channel is provided in the paper. Two parallel BP decoders, one employing the standard graph and the other selected based on the selection principle, are also proposed to work together. The simulation results show that when the total number of iterations is fixed, the proposed BP decoding can have the same performance as the existing permutation scheme while provides a higher throughput.

IEEE Access•Jan 1, 2020•8 citations
Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve the capacity of symmetric binary-input memoryless channels. Here, we study the bit error rate performance of finite-length polar codes under Belief Propagation (BP) decoding. We analyze the stopping sets of polar codes and the size of the minimal stopping set, called “stopping distance”. Stopping sets, as they contribute to the decoding failure, play an important role in bit error rate and error floor performance of the code. We show that the stopping distance for binary polar codes, if carefully designed, grows as O(√N) where N is the code-length. We provide bit error rate (BER) simulations for polar codes over binary erasure and gaussian channels, showing no sign of error floor down to the BERs of 10-11. Our simulations asserts that while finite-length polar codes do not perform as good as LDPC codes in terms of bit error rate, they show superior error floor performance. Motivated by good error floor performance, we introduce a modified version of BP decoding employing a guessing algorithm to improve the BER performance of polar codes. Our simulations for this guessing algorithm show two orders of magnitude improvement over simple BP decoding for the binary erasure channel (BEC), and up to 0.3 dB improvement for the gaussian channel at BERs of 10-6.
In this paper, we discuss the belief propagation decoding algorithm for polar codes. The performance of Polar codes for shorter lengths is not satisfactory. Motivated by this, we propose a novel technique to improve its performance at short lengths. We showed that the probability of messages passed along the factor graph of polar codes, can be increased by multiplying the current message of nodes with their previous message. This is like a feedback path in which the present signal is updated by multiplying with its previous signal. Thus the experimental results show that performance of belief propagation polar decoder can be improved using this proposed technique. Simulation results in binary-input additive white Gaussian noise channel (BI-AWGNC) show that the proposed belief propagation polar decoder can provide significant gain of 2 dB over the original belief propagation polar decoder with code rate 0.5 and code length 128 at the bit error rate (BER) of 10−410^{-4}10−4 .
Polar Codes are a new class of capacity-reaching codes that have low decoding complexity and excessive reliability. They are primarily based on the concept of channel polarization, which makes a few bits inside the codeword more reliable and a few much less reliable because the code duration increases. This examination comprehensively evaluates how effective Belief Propagation (BP) and Belief Propagation with List (BPL) Decoding algorithms are in the Binary Erasure Channel (BEC) setup. The fundamental purpose is to recognize how nicely these algorithms deal with errors and enhance the reliability of verbal exchange structures. Different code lengths and list sizes are examined to see how they affect the Block Error Rate (BLER). This paper considers 4 codes studied and carried out in the usage of MATLAB lengths: N=256, 512, 1024, and 2048, and varies the Energy per bit to Noise power (Eb/No) from 1 dB to a 3.5 dB with list sizes eight and sixteen. The experiments show how the BP and BPL algorithms perform in coping with erasure errors and ensuring reliable deciphering. The BLER figures provide an in-depth description and comparison of the BP and BPL interpreting algorithms. The analysis highlights the promising blunder-correction talents of each BP and BPL algorithm in the BEC context. As the code length increases, the capability to correct errors also improves, showing that these algorithms can scale effectively. The addition of list decoding in BPL similarly enhances blunder correction by offering multiple possible answers. These examine the effect of Eb/No stages on the performance of BP and BPL deciphering, displaying that better Eb/No stages enhance accuracy but also increase the erasure chance. This paper discusses the trade-offs between accuracy and complexity for BP and BPL deciphering, presenting insights into their performance traits and optimization.
In the construction of polar code, the selection of frozen bits affects the error-correcting performance significantly. Several calculation-based algorithms have been proposed for general binary-input discrete memoryless channels (B-DMCs) like the additive white Gaussian noise (AWGN) channel. In this paper, a method for frozen bits selection based on Monte Carlo simulation and belief propagation (BP) decoding is proposed. The information bits are selected out one by one incrementally. The numerical results show that the proposed algorithm can effectively improve the bit error rate (BER) and frame error rate (FER) performance compared with the conventional selection method, especially in high signal noise ratio (SNR) region. Moreover, the algorithm can be used to construct polar codes with any rate through a complete iteration.
This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation (BP) decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of stopping sets. In the first part we describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or until the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum size turbo stopping set which is also a subset of the set of erased positions. In the second part we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results
Assuming iterative decoding for binary erasure channels (BECs), a novel tree-based technique for upper bounding the bit error rates (BERs) of arbitrary, finite low-density parity-check (LDPC) codes is provided and the resulting bound can be evaluated for all operating erasure probabilities, including both the waterfall and the error floor regions. This upper bound can also be viewed as a narrowing search of stopping sets, which is an approach different from the stopping set enumeration used for lower bounding the error floor. When combined with optimal leaf-finding modules, this upper bound is guaranteed to be tight in terms of the asymptotic order. The Boolean framework proposed herein further admits a composite search for even tighter results. For comparison, a refinement of the algorithm is capable of exhausting all stopping sets of size les 13 for irregular LDPC codes of length n ap 500, which requires (13500) ap 1.67 times 1025trials if a brute force approach is taken. These experiments indicate that this upper bound can be used both as an analytical tool and as a deterministic worst-performance (error floor) guarantee, the latter of which is crucial to optimizing LDPC codes for extremely low BER applications, e.g., optical/satellite communications
The belief propagation (BP) decoding of polar codes provides better bit error rate (BER) and packet error rate (PER) performance than the successive cancelation (SC) decoding. It has been reported in the literature that the graph of BP decoding of polar codes can be permuted to provide even better decoding performance. In this paper, we theoretically prove that all permutations of the graph are equivalent in terms of the polar encoding. For BP decoding, it is shown that permutations of the layers of the graph also permute the bit channels. A principle which protects the weakest information bit channel is provided in the paper. Two parallel BP decoders, one employing the standard graph and the other selected based on the selection principle, are also proposed to work together. The simulation results show that when the total number of iterations is fixed, the proposed BP decoding can have the same performance as the existing permutation scheme while provides a higher throughput.