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arXiv:2112.10735

Successive Cancellation Ordered Search Decoding of Modified $\boldsymbol{G}_N$-Coset Codes

Peihong Yuan, Mustafa Cemil Coşkun

Published Dec 20, 2021• 0 citations
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Abstract

A tree search algorithm called successive cancellation ordered search (SCOS) is proposed for $\boldsymbol{G}_N$-coset codes that implements maximum-likelihood (ML) decoding with adaptive complexity for transmission over binary-input AWGN channels. Unlike bit-flip decoders, no outer code is needed to terminate decoding; therefore, SCOS also applies to $\boldsymbol{G}_N$-coset codes modified with dynamic frozen bits. The average complexity is close to that of successive cancellation (SC) decoding at practical frame error rates (FERs) for codes with wide ranges of rate and lengths up to $512$ bits, which perform within $0.25$ dB or less from the random coding union bound and outperform Reed--Muller codes under ML decoding by up to $0.5$ dB. Simulations illustrate simultaneous gains for SCOS over SC-Fano, SC stack (SCS) and SC list (SCL) decoding in FER and the average complexity at various SNR regimes. SCOS is further extended by forcing it to look for candidates satisfying a threshold, thereby outperforming basic SCOS under complexity constraints. The modified SCOS enables strong error-detection capability without the need for an outer code. In particular, the $(128, 64)$ polarization-adjusted convolutional code under modified SCOS provides gains in overall and undetected FER compared to CRC-aided polar codes under SCL/dynamic SC flip decoding at high SNR.

DOI
10.48550/arxiv.2112.10735

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Successive Cancellation Ordered Search Decoding of Modified GN\boldsymbol{G}_NGN​-Coset Codes
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Complexity-Adaptive Maximum-Likelihood Decoding of Modified GN\boldsymbol{G}_NGN​-Coset Codes
A complexity-adaptive tree search algorithm is proposed for GN\boldsymbol{G}_NGN​-coset codes that implements maximum-likelihood (ML) decoding by using a successive decoding schedule. The average complexity is close to that of the successive cancellation (SC) decoding for practical error rates when applied to polar codes and short Reed-Muller (RM) codes, e.g., block lengths up to N=128N=128N=128. By modifying the algorithm to limit the worst-case complexity, one obtains a near-ML decoder for longer RM codes and their subcodes. Unlike other bit-flip decoders, no outer code is needed to terminate decoding. The algorithm can thus be applied to modified GN\boldsymbol{G}_NGN​-coset code constructions with dynamic frozen bits. One advantage over sequential decoders is that there is no need to optimize a separate parameter.
A complexity-adaptive tree search algorithm is proposed for GNG_{N}GN​-coset codes that implements maximum-likelihood (ML) decoding by using a successive decoding schedule. The average complexity is close to that of the successive cancellation (SC) decoding for practical error rates when applied to polar codes and short Reed-Muller (RM) codes, e.g., block lengths up to N = 128. By modifying the algorithm to limit the worstcase complexity, one obtains a near-ML decoder for longer RM codes and their subcodes. Unlike other bit-flip decoders, no outer code is needed to terminate decoding. The algorithm can thus be applied to modified GNG_{N}GN​-coset code constructions with dynamic frozen bits. One advantage over sequential decoders is that there is no need to optimize a separate parameter.
As two common successive cancellation (SC)-based decoding algorithms of polar codes, the SC list (SCL) and SC stack (SCS) decoder can achieve satisfactory error correction performance, especially with increased list size or stack depth. Nevertheless, a large list size or stack depth will lead to high computational complexities and hardware resources. To this end, successive cancellation priority (SCP) decoding with priority- first searching strategy and trellis-like storage is proposed to offer one solution. In this paper, an efficient SCP decoder is first proposed to verify its advantages over SCL and SCS decoders. Furthermore, an adaptive node-inserting scheme is proposed to reduce the number of bits insert into the priority queue. Numerical results have shown that for the polar code with transmission length 1024 and rate 1/2, the proposed adaptive SCP (ASCP) decoder can achieve significant time complexity reduction on average compared with the standard SCL decoder. The hardware architecture of SCP decoding is implemented using 65-nm CMOS technology and the results show better throughput compared with the SCS decoder.
Polar Codes achieve average error correction performance at finite code lengths, when decoded using Successive Cancellation (SC) decoding. Successive Cancellation Flip (SC-Flip) based decoders provide a viable low complexity substitute of the high performing high complexity Successive Cancellation List (SCL) decoders, providing a considerable improvement in the performance. A novel approach to identify and flip multiple erroneous bits in each decoding attempt has been aimed in this paper. Moreover, a faster version of the proposed decoder is also presented, which provides better performance throughout the entire decoding range. The performance of the proposed decoders have been evaluated with respect to frame error correction rate along with average normalized complexity and then compared with some of the existing works for validating the outperformance.
Polar codes are the first error-correcting codes that achieve the Shannon limit under successive cancellation (SC) decoding when the code length is infinite. However, the error-correcting performance of SC decoding is mediocre for short and moderate code lengths. Dynamic successive cancellation flip (DSCF) decoding has been proposed that performs extra SC decoding trails with multi-bit flipping to improve the performance of short and moderate polar codes. The DSCF decoding suffers from high computational complexity and decoding latency since all information bits are tested when searching for the flipping candidates, which leads to intensive metric calculations. To reduce the search complexity of the DSCF, a method based on the critical set (CS) is proposed in this paper to narrow the search range and two types of special nodes, rate-0 nodes and repetition nodes, are incorporated for further complexity reduction. Simulation results show that the proposed method can achieve more than 70% reduction in calculations on metrics, and up to 22.79% reduction in calculations on nodes without performance loss compared to the original DSCF decoding.
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.