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Decoding of Polar Codes with Future Constraints Over the Binary Erasure Channel

Min Jang, Jong-Hwan Kim, Seho Myung, Kyeongcheol Yang

Published Dec 4, 2023• 0 citations
Publisher

Abstract

We address the suboptimality of successive cancellation (SC) decoding for polar codes, which is caused by treating as random variables all the future bits to be estimated later. For a target information bit, we define the frozen and parity bits located behind it as its future constraints (FCs). To incorporate FCs into the sequential decoding of the target bit, we propose two elementary techniques: an SC check (SCC) algorithm and an FC conversion rule enabling belief propagation (BP). Focusing on the binary erasure channel (BEC), we also present a tree search technique based on stack-based backjumping (SBJ) to efficiently solve dynamic constraint satisfaction problems (CSP) formulated by FCs. Numerical results show that the combination of BP and SCC decoding algorithms accompanied with the SBJ technique achieves excellent erasure recovery performance over the BEC, which is close to the dependence testing (DT) achievability bound.

DOI
10.1109/globecom54140.2023.10437704

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We address the suboptimality of successive cancellation (SC) decoding for polar codes, which is caused by treating as random variables all the future bits to be estimated later. For a target information bit, we define the frozen and parity bits located behind it as its future constraints (FCs). To incorporate FCs into the sequential decoding of the target bit, we propose two elementary techniques: an SC check (SCC) algorithm and an FC conversion rule enabling belief propagation (BP). Focusing on the binary erasure channel (BEC), we also present a tree search technique based on stack-based backjumping (SBJ) to efficiently solve dynamic constraint satisfaction problems (CSP) formulated by FCs. Numerical results show that the combination of BP and SCC decoding algorithms accompanied with the SBJ technique achieves excellent erasure recovery performance over the BEC, which is close to the dependence testing (DT) achievability bound.
In the conventional successive cancellation (SC) decoder for polar codes, all the future bits to be estimated later are treated as random variables. However, polar codes inevitably involve frozen bits, and their concatenated coding schemes also include parity bits (or dynamic frozen bits) causally generated from the past bits estimated earlier. We refer to the frozen and parity bits located behind a target decoding bit as its future constraints (FCs). Although the values of FCs are deterministic given the past estimates, they have not been exploited in the conventional SC-based decoders, not leading to optimality. In this paper, with a primary focus on the binary erasure channel (BEC), we propose SC-check (SCC) and belief propagation SCC (BP-SCC) decoding algorithms in order to leverage FCs in decoding. We further devise an improved tree search technique based on stack-based backjumping (SBJ) to solve dynamic constraint satisfaction problems (CSPs) formulated by FCs. Over the BEC, numerical results show that a combination of the BP-SCC algorithm and the SBJ tree search technique achieves the erasure recovery performance close to the dependence testing (DT) bound, a bound of achievable finite-length performance.
Polar codes are the first codes to provably achieve capacity on the symmetric binary-input discrete memoryless channel (B-DMC) with low encoding and decoding complexity. The parity check matrix of polar codes is high-density and we show that linear program (LP) decoding fails on the fundamental polytope of the parity check matrix. The recursive structure of the code permits a sparse factor graph representation. We define a new polytope based on the fundamental polytope of the sparse graph representation. This new polytope P is defined in a space of dimension O(N logN) where N is the block length. We prove that the projection of P in the original space is tighter than the fundamental polytope based on the parity check matrix. The LP decoder over P obtains the ML-certificate property. In the case of the binary erasure channel (BEC), the new LP decoder is equivalent to the belief propagation (BP) decoder operating on the sparse factor graph representation, and hence achieves capacity. Simulation results of SC (successive cancellation) decoding, LP decoding over tightened polytopes, and (ML) maximum likelihood decoding are provided. For channels other than the BEC, we discuss why LP decoding over P with a linear objective function is insufficient.
The bit-flip method has been successfully applied to the successive cancellation (SC) decoder to improve the block error rate (BLER) performance for polar codes in the finite code length region. However, due to the sequential decoding, the SC decoder inherently suffers from longer decoding latency than that of the belief propagation (BP) decoder with efficient early stopping criterion. It is natural to ask how to perform bit-flip in a polar BP decoder. In this paper, bit-flip is introduced into the BP decoder for polar codes. The idea of critical set (CS), that is, originally proposed by Zhang et al. for identifying unreliable bits in a SC bit-flip decoder, is extended to the BP decoder here. After revealing the relationship between CS and the incorrect BP decoding results, critical set with order ω (CS-ω) is constructed to identify unreliable bit decisions in polar BP decoding. The simulation results demonstrate that compared with the conventional BP decoder, the BLER of the proposed bit-flip decoder can achieve significant signal-to-noise ratio (SNR) gain which is comparable to that of a cyclic redundancy check-aided SC list decoder with a moderate list size. In addition, the decoding latency of the proposed BP bit-flip decoder is only slightly higher than that of the conventional BP decoder in the medium and high SNR regions.
We introduce successive cancellation (SC) decoding of product codes (PCs) with single parity-check (SPC) component codes. Recursive formulas are derived, which resemble the SC decoding algorithm of polar codes. We analyze the error probability of SPC-PCs over the binary erasure channel under SC decoding. A bridge with the analysis of PCs introduced by Elias in 1954 is also established. Furthermore, bounds on the block error probability under SC decoding are provided, and compared to the bounds under the original decoding algorithm proposed by Elias. It is shown that SC decoding of SPC-PCs achieves a lower block error probability than Elias' decoding.
Polar codes represent an emerging correcting codes with power to approach the capacity of a discrete memoryless channel.However, the decoding performance of polar codes with traditional successive cancellation (SC) algorithm cannot match that of the low-density parity-check or Turbo codes.In this paper, a modified successive cancellation stack (MSCS) polar decoder is proposed.Unlike the original SC stack polar (SCS) decoders, we propose a modified path that can significantly reduce latency of SCS decoders.Simulation results in the binary-input additive white Gaussian noise channel show that modified algorithm has the slightly performance as the SCS algorithm.Moreover, the time and space complexity of the MSCS decoder is much lower than that of the SCS and can be very close to that of the SC in the high SNR regime.
A successive cancellation (SC) decoder with inactivations is proposed as an efficient implementation of SC list (SCL) decoding over the binary erasure channel. The proposed decoder assigns a dummy variable to an information bit whenever it is erased during SC decoding and continues with decoding. Inactivated bits are resolved using information gathered from decoding frozen bits. This decoder leverages the structure of the Hadamard matrix, but can be applied to any linear code by representing it as a polar code with dynamic frozen bits. SCL decoders are partially characterized using density evolution to compute the average number of inactivations required to achieve the maximum a-posteriori decoding performance. The proposed measure quantifies the performance vs. complexity trade-off and provides new insight into dynamics of the number of paths in SCL decoding. The technique is applied to analyze Reed-Muller (RM) codes with dynamic frozen bits. It is shown that these modified RM codes perform close to extended BCH codes.