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arXiv:2102.07362 · IEEE International Symposium on Information Theory (ISIT)

A Deterministic Algorithm for Computing the Weight Distribution of Polar Codes

Hanwen Yao, Arman Fazeli, Alexander Vardy

Published Jul 12, 2021• 26 citations
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Abstract

We present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distributions that arise in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, we can compute the entire weight distribution of certain polar cosets in time$O(n^{2})$. Any polar code can be represented as a disjoint union of such cosets; moreover, this representation extends to polar codes with dynamically frozen bits. This implies that our methods can be also used to compute the weight distribution of polar codes with CRC precoding, of pol-arization-adjusted convolutional (PAC) codes and, in fact, general linear codes. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To reduce the exponential complexity of our algorithm, we make use of the fact that polar codes have a large automorphism group, which includes the lower-triangular affine group LTA$(m,2)$. We prove that LTA$(m,2)$acts transitively on certain subsets of polar codes, thereby drastically reducing the number of polar cosets we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of any polar code of length up to$n=128$.

DOI
10.1109/isit45174.2021.9517950

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We present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distributions that arise in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, we can compute the entire weight distribution of certain polar cosets in timeO(n2)O(n^{2})O(n2). Any polar code can be represented as a disjoint union of such cosets; moreover, this representation extends to polar codes with dynamically frozen bits. This implies that our methods can be also used to compute the weight distribution of polar codes with CRC precoding, of pol-arization-adjusted convolutional (PAC) codes and, in fact, general linear codes. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To reduce the exponential complexity of our algorithm, we make use of the fact that polar codes have a large automorphism group, which includes the lower-triangular affine group LTA(m,2)(m,2)(m,2). We prove that LTA(m,2)(m,2)(m,2)acts transitively on certain subsets of polar codes, thereby drastically reducing the number of polar cosets we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of any polar code of length up ton=128n=128n=128.
In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code lengthn, we can compute the weight distribution of anypolar cosetsin timeO(n2). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call themixing factor. To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at lengthn= 128.
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.
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 constructed with large polarization kernels were recently proven to be able to achieve optimal finite-length scaling properties. However, straightforward decoding for large kernel polar codes introduces a complexity coefficient that is exponential to the kernel sizes, which makes such codes generally believed to be impractical. In this paper, we present a new method that decodes large kernel polar codes with a complexity coefficient that is polynomial to the kernel sizes. This could facilitate the implementation of large kernel polar codes for practical use in the future.Successive cancellation decoding for large kernel polar codes requires calculation on the probabilities of its bit channels. Similar to conventional polar codes, those bit channels follow a recursive relation, which make this calculation boil down to computing the probabilities for bit channels of a single polarization kernel. This kernel-level computation can be shown equivalent to soft-output maximum-likelihood (ML) decoding on a single bit of a linear block code. In our proposed method, we first use linear operations to represent the considered linear block code as a polar code with dynamically frozen bits, and then use a modified polar list decoder to get an approximate value on the soft-output of the desired bit. This method is motivated by the observation that at short block lengths, polar list decoding with a large enough list size can well-approximate ML decoding. The proposed low-complexity method allows us to decode polar codes constructed with a 64 × 64 polarization kernel with scaling exponent µ ≈ 2.87 for the first time.
In this paper, we derive the exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes. Though we do not compute the distance spectrum of polar codes, the results allow us to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels. Also, we establish the minimal distance between two cosets associated with two paths that differ in two positions. This can be regarded as a first step toward analyzing the weight distributions for successive cancellation list decoding.