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

IEEE Communications Letters

On Probabilistic Weight Distribution of Polar Codes

Mehrdad Valipour, Shahram Yousefi

Published Sep 16, 2013• 35 citations
Publisher

Abstract

Polar code is a low-complexity and capacity-achieving code with applications in channel coding, source coding and secrecy. One way to evaluate the error performance of the finite-length polar 'channel' code, for instance, is to use Bonferroni-type (e.g., union bound) bounds requiring the Hamming weight distribution. Currently, there are no low-complexity algorithms for the polar code's spectra. In this paper, we propose a recursive probabilistic weight distribution (PWD) expression for polar codes. As this expression is complex, two polynomial-time approximations and a tree-based approach have been presented for a faster implementation of the PWD method with much accuracy.

DOI
10.1109/lcomm.2013.091113.131213

Related papers

MacWilliams Identities for Probabilistic Weight Distribution of Polar Codes

Fengyi Cheng, Aijun Liu, Biao Cai +1 more

The problem of estimating the probabilistic Hamming weight distribution (HWD) is consider for polar codes. Two previous computationally recursive algorithm are reviewed and extended to the whole rate range. For low code rate, the imprecise assumption of previous works can be rectified by calculating HWD of the corresponding dual code. Based on MacWilliams Identities, the HWD of polar codes can be enumerated. By regarding as concatenation of two subcodes, the high rate one and low rate one, the HWD of medium code rate can be derived. The estimated result yields much accurate asymptotic performance bound.

International Conference on Information Science and Control Engineering•Dec 1, 2019•1 citation

A Parity-Consistent Decomposition Method to Determine the Weight Distribution of Polar Codes

Ying Liu, Bolin Wu, Yuxin Han +1 more

This paper introduces a novel parity-consistent decomposition (PCD) method for accurately determining the weight distribution (WD) of polar codes. In the PCD method, polar codes are represented as a union of a new type of polar cosets, characterized by the property that any two adjacent bits in the uncoded sequence are either both information bits or both frozen bits. This representation reduces the number of polar cosets compared to the existing automorphism-based (AUT) method. We introduce a recursive decomposition method for this new polar coset, which facilitates efficient recursive computation of the WD of polar codes. The PCD method is organized within a tree structure, which facilitates the evaluation of its computational complexity. We then investigate the transitivity of the action of lower triangular affine (LTA) transformations on a code set composed of the new polar cosets. This transitivity property enables a further reduction in the number of cosets that need to be computed. Furthermore, we propose a pruning algorithm to determine the partial weight distribution (PWD) of polar codes based on the tree structure of the PCD method. Given an upper bound on the Hamming weight for the PWD, we update the Hamming weight threshold for each code in the tree and remove the codes whose minimum Hamming weights exceed their corresponding thresholds. Numerical results demonstrate the superiority of the PCD method over existing methods.

IEEE Transactions on Communications•Jan 1, 2026•1 citation

Approximate Weight Distribution of Polarization-Adjusted Convolutional (PAC) Codes

Sadra Seyedmasoumian, Tolga M. Duman

Polarization-adjusted convolutional (PAC) codes combine polar and convolutional transformations to enhance the distance properties of polar codes. They offer a performance very close to the finite-length information-theoretic bounds for short blocklengths. In this paper, we develop a method of computing the weight distribution of PAC codes in an approximate form by employing a probabilistic technique. We demonstrate that the results well match the exact weight distributions for small codes that can be computed using a brute-force algorithm. We also present a way of employing the results (along with a union bound on the code performance) to design specific PAC codes or, more precisely, to determine suitable rate profiles via simulated annealing. Numerical examples illustrate that the PAC codes with the designed rate profiles offer superior performance.

arXiv:2202.12885•IEEE International Symposium on Information Theory (ISIT)•Jun 26, 2022•15 citations

A Parity-Consistent Decomposition Algorithm to Determine the Weight Distribution of Polar Codes

Yang Liu, Bolin Wu, Yuxin Han +1 more

This paper introduces a novel parity-consistent decomposition (PCD) algorithm for accurately determining the weight distribution (WD) of polar codes. In the PCD algorithm, polar codes are represented as a union of a new type of polar cosets, which avoids fixing two adjacent information bits as 0 and 1. This representation reduces the number of polar cosets compared to the existing automorphism-based algorithm. Furthermore, we introduce a recursive decomposition method for this new polar coset, which facilitates efficient recursive computation of the WD of polar codes. The PCD algorithm is organized within a tree-like structure, which facilitates the evaluation of its computational complexity. Comparative analysis with the automorphism-based method demonstrates that the PCD algorithm significantly reduces computational complexity.

IEEE International Symposium on Information Theory (ISIT)•Jun 22, 2025•3 citations

Evaluate PAC codes via Efficient Estimation on Weight Distribution

Junhua You, Shaohua Wu, Yajing Deng +1 more

In this letter, we introduce an efficient method for estimating weight distributions of polar codes and polarization-adjusted convolutional (PAC) codes. Based on a recursive algorithm of computing the weight enumerating functions of polar cosets, this method focuses on two key objectives: accurately determining the number of low-weight codewords and quickly approximating the distribution of high-weight codewords. Simulation results demonstrate that this hybrid method maintains competitively low complexity while effectively achieving the objectives.

arXiv:2310.08226•Oct 12, 2023•0 citations

A Lower Bound on the Probability of Error of Polar Codes over BMS Channels

Boaz Shuval, Ido Tal

Polar codes are a family of capacity-achieving codes that have explicit and low-complexity construction, encoding, and decoding algorithms. Decoding of polar codes is based on the successive-cancellation decoder, which decodes in a bit- wise manner. A decoding error occurs when at least one bit is erroneously decoded. The various codeword bits are correlated, yet performance analysis of polar codes ignores this dependence: the upper bound is based on the union bound, and the lower bound is based on the worst-performing bit. Improvement of the lower bound is afforded by considering error probabilities of two bits simultaneously. These are difficult to compute explicitly due to the large alphabet size inherent to polar codes. In this research we propose a method to lower-bound the error probabilities of bit pairs. We develop several transformations on pairs of synthetic channels that make the resultant synthetic channels amenable to alphabet reduction. Our method yields lower bounds that significantly improve upon currently known lower bounds for polar codes under successive-cancellation decoding.

arXiv:1701.01628•Jan 6, 2017•0 citations
Polar code is a low-complexity and capacity-achieving code with applications in channel coding, source coding and secrecy. One way to evaluate the error performance of the finite-length polar 'channel' code, for instance, is to use Bonferroni-type (e.g., union bound) bounds requiring the Hamming weight distribution. Currently, there are no low-complexity algorithms for the polar code's spectra. In this paper, we propose a recursive probabilistic weight distribution (PWD) expression for polar codes. As this expression is complex, two polynomial-time approximations and a tree-based approach have been presented for a faster implementation of the PWD method with much accuracy.
The problem of estimating the probabilistic Hamming weight distribution (HWD) is consider for polar codes. Two previous computationally recursive algorithm are reviewed and extended to the whole rate range. For low code rate, the imprecise assumption of previous works can be rectified by calculating HWD of the corresponding dual code. Based on MacWilliams Identities, the HWD of polar codes can be enumerated. By regarding as concatenation of two subcodes, the high rate one and low rate one, the HWD of medium code rate can be derived. The estimated result yields much accurate asymptotic performance bound.
This paper introduces a novel parity-consistent decomposition (PCD) method for accurately determining the weight distribution (WD) of polar codes. In the PCD method, polar codes are represented as a union of a new type of polar cosets, characterized by the property that any two adjacent bits in the uncoded sequence are either both information bits or both frozen bits. This representation reduces the number of polar cosets compared to the existing automorphism-based (AUT) method. We introduce a recursive decomposition method for this new polar coset, which facilitates efficient recursive computation of the WD of polar codes. The PCD method is organized within a tree structure, which facilitates the evaluation of its computational complexity. We then investigate the transitivity of the action of lower triangular affine (LTA) transformations on a code set composed of the new polar cosets. This transitivity property enables a further reduction in the number of cosets that need to be computed. Furthermore, we propose a pruning algorithm to determine the partial weight distribution (PWD) of polar codes based on the tree structure of the PCD method. Given an upper bound on the Hamming weight for the PWD, we update the Hamming weight threshold for each code in the tree and remove the codes whose minimum Hamming weights exceed their corresponding thresholds. Numerical results demonstrate the superiority of the PCD method over existing methods.
Polarization-adjusted convolutional (PAC) codes combine polar and convolutional transformations to enhance the distance properties of polar codes. They offer a performance very close to the finite-length information-theoretic bounds for short blocklengths. In this paper, we develop a method of computing the weight distribution of PAC codes in an approximate form by employing a probabilistic technique. We demonstrate that the results well match the exact weight distributions for small codes that can be computed using a brute-force algorithm. We also present a way of employing the results (along with a union bound on the code performance) to design specific PAC codes or, more precisely, to determine suitable rate profiles via simulated annealing. Numerical examples illustrate that the PAC codes with the designed rate profiles offer superior performance.
This paper introduces a novel parity-consistent decomposition (PCD) algorithm for accurately determining the weight distribution (WD) of polar codes. In the PCD algorithm, polar codes are represented as a union of a new type of polar cosets, which avoids fixing two adjacent information bits as 0 and 1. This representation reduces the number of polar cosets compared to the existing automorphism-based algorithm. Furthermore, we introduce a recursive decomposition method for this new polar coset, which facilitates efficient recursive computation of the WD of polar codes. The PCD algorithm is organized within a tree-like structure, which facilitates the evaluation of its computational complexity. Comparative analysis with the automorphism-based method demonstrates that the PCD algorithm significantly reduces computational complexity.
In this letter, we introduce an efficient method for estimating weight distributions of polar codes and polarization-adjusted convolutional (PAC) codes. Based on a recursive algorithm of computing the weight enumerating functions of polar cosets, this method focuses on two key objectives: accurately determining the number of low-weight codewords and quickly approximating the distribution of high-weight codewords. Simulation results demonstrate that this hybrid method maintains competitively low complexity while effectively achieving the objectives.
Polar codes are a family of capacity-achieving codes that have explicit and low-complexity construction, encoding, and decoding algorithms. Decoding of polar codes is based on the successive-cancellation decoder, which decodes in a bit- wise manner. A decoding error occurs when at least one bit is erroneously decoded. The various codeword bits are correlated, yet performance analysis of polar codes ignores this dependence: the upper bound is based on the union bound, and the lower bound is based on the worst-performing bit. Improvement of the lower bound is afforded by considering error probabilities of two bits simultaneously. These are difficult to compute explicitly due to the large alphabet size inherent to polar codes. In this research we propose a method to lower-bound the error probabilities of bit pairs. We develop several transformations on pairs of synthetic channels that make the resultant synthetic channels amenable to alphabet reduction. Our method yields lower bounds that significantly improve upon currently known lower bounds for polar codes under successive-cancellation decoding.