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IEEE International Symposium on Information Theory (ISIT)

On the Average Weight Spectrum of Pre-Transformed Rate-Compatible Polar Codes

Yuan Li, Zicheng Ye, Huazi Zhang, Jun Wang, Guiying Yan, Zhiming Ma

Published Jun 22, 2025• 0 citations
Publisher

Abstract

The weight spectrum plays a crucial role in the performance of error-correcting codes. Pre-transformation with an upper-triangular matrix improves the weight spectrum of polar codes while retaining polarization. However, a theoretical analysis to quantify the improvement for pre-transformed rate-compatible polar codes is missing. In this paper, we calculate the average spectrum of random upper-triangular pre-transformed shortened and punctured polar codes. Our approach tran-scends the limitations imposed by partial ordering and specific rate matching patterns. A key feature of our approach is its polynomial complexity in relation to the code length, making it computationally feasible. Simulation results affirm that our findings provide an accurate approximation on the performance of pre-transformed rate-compatible polar codes.

DOI
10.1109/isit63088.2025.11195397

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The weight spectrum plays a crucial role in the performance of error-correcting codes. Pre-transformation with an upper-triangular matrix improves the weight spectrum of polar codes while retaining polarization. However, a theoretical analysis to quantify the improvement for pre-transformed rate-compatible polar codes is missing. In this paper, we calculate the average spectrum of random upper-triangular pre-transformed shortened and punctured polar codes. Our approach tran-scends the limitations imposed by partial ordering and specific rate matching patterns. A key feature of our approach is its polynomial complexity in relation to the code length, making it computationally feasible. Simulation results affirm that our findings provide an accurate approximation on the performance of pre-transformed rate-compatible polar codes.
The weight spectrum plays a crucial role in the performance of error-correcting codes. Despite substantial theoretical exploration into polar codes with mother code length, a framework for the weight spectrum of rate-compatible polar codes remains elusive. In this paper, we address this gap by enumerating the number of minimum-weight codewords for quasi-uniform punctured, Wang-Liu shortened, and bit-reversal shortened decreasing polar codes. Notably, our algorithms operate with polynomial complexity relative to the code length. Simulation results affirm that our discoveries provide an accurate approximation on the performance of rate-compatible polar codes.
Pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and polarization-adjusted convolutional (PAC) codes) improves the weight spectrum of Reed-Muller (RM) codes and polar codes significantly. However, a theoretical analysis to quantify the improvement is missing. In this paper, we provide asymptotic analysis on the number of low-weight codewords of the original and pre-transformed RM codes respectively, and prove that pre-transformation significantly reduces low-weight codewords, even in the order sense. For polar codes, we prove that the average number of minimum-weight codewords does not increase after pre-transformation. Both results confirm the advantages of pre-transformation.
In this paper, we present a deterministic algorithm to count the low-weight codewords of punctured and shortened pure and pre-transformed polar codes. The method first evaluates the weight properties of punctured/shortened polar cosets. Then, a method that discards the cosets that have no impact on the computation of the low-weight codewords is introduced. A key advantage of this method is its applicability, regardless of the frozen bit set, puncturing/shortening pattern, or pre-transformation. Results confirm the method’s efficiency while showing reduced computational complexity compared to state-of-the-art algorithms.
The minimum Hamming distance of a linear block code is the smallest number of bit changes required to transform one valid codeword into another. The code’s minimum distance determines the code’s error-correcting capabilities. Furthermore, The number of minimum weight codewords, a.k.a. error coefficient, gives a good comparative measure for the block error rate (BLER) of linear block codes with identical minimum distance, in particular at a high SNR regime under maximum likelihood (ML) decoding. A code with a smaller error coefficient would give a lower BLER. Unlike polar codes, a closed-form expression for the enumeration of the error coefficient of polarization-adjusted convolutional (PAC) codes is yet unknown. As PAC codes are convolutionally pre-transformed polar codes, we study the impact of pre-transformation on polar codes in terms of minimum Hamming distance and error coefficient by partitioning the codewords into cosets. We show that the minimum distance of PAC codes does not decrease; however, the pre-transformation may reduce the error coefficient depending on the choice of convolutional polynomial. We recognize the properties of the cosets where pre-transformation is ineffective in decreasing the error coefficient, giving a lower bound for the error coefficient. Then, we propose a low-complexity enumeration method that determines the number of minimum weight codewords of PAC codes relying on the error coefficient of polar codes. That is, given the error coefficient Awmin of polar codes, we determine the reduction X in the error coefficient due to convolutional pre-transformation in PAC coding and subtract it from the error coefficient of polar codes, Awmin - X. Furthermore, we numerically analyze the tightness of the lower bound and the impact of the choice of the convolutional polynomial on the error coefficient based on the sub-patterns in the polynomial’s coefficients. Eventually, we show how we can further reduce the error coefficient in the cosets.
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.
In this paper, we put forth a novel class of pre-transformed polar codes, called as “deep-polar codes.” Encoding of deep-polar codes involves a series of multi-layered polar transformations. By applying these successive polar-based pre-transformations, deep-polar codes with an improved weight distribution are constructed with low encoding complexity. We also propose a low-complexity successive cancellation list decoding algorithm to effectively decode the deep-polar codes. This decoding algorithm harnesses the reverse process of the multi-layered pre-transformed encoding, referred to as “back propagation parity checks.” From simulations, we demonstrate that deep-polar codes outperform existing pre-transformed polar codes in terms of block error rates across various code rates and short blocklengths, while maintaining low-complexity encoding and decoding. Our findings highlight the effectiveness and practicality of deep-polar codes for short blocklength transmission scenarios in 6G.