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) 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.
This paper introduces an efficient algorithm based on the Parity-Consistent Decomposition (PCD) method to determine the WD of pre-transformed polar codes. First, to address the bit dependencies introduced by the pre-transformation matrix, we propose an iterative algorithm to construct an \emph{Expanded Information Set}. By expanding the information bits within this set into 0s and 1s, we eliminate the correlations among information bits, thereby enabling the recursive calculation of the Hamming weight distribution using the \emph{PCD method}. Second, to further reduce computational complexity, we establish the theory of equivalence classes for pre-transformed polar codes. Codes within the same equivalence class share an identical weight distribution but correspond to different \emph{Expanded Information Set} sizes. By selecting the pre-transformation matrix that minimizes the \emph{Expanded Information Set} size within an equivalence class, we optimize the computation process. Numerical results demonstrate that the proposed method significantly reduces computational complexity compared to existing deterministic algorithms.
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.
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.
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). 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 ton=128.
The problem of computing the Hamming weight distribution of linear codes is considered in this paper. A novel method to enumerate all codewords up to a certain Hamming weight for binary linear block codes in general and in particular for the punctured, shortened, precoded polar codes is introduced. The proposed approach performs a recursive decomposition of the codes using construction X4 that is typically used to combine codes of different lengths. This allows to enumerate the low-weight codewords of the overall code as combinations of the low-weight codewords of the component codes. Numerical results show that the proposed approach can efficiently compute the exact partial weight distribution of the 5G New Radio punctured/shortened polar codes with CRC11 and pure polar codes. In the former and latter cases, the low-weight codeword number is up to 106 and 108, respectively. Besides, randomly punctured and shortened polar codes and randomly precoded polar codes are also considered. To the best of the authors’ knowledge, this is the first method able to solve these problems.