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

TL;DR

The paper introduces SCOS, a tree-search maximum-likelihood decoder for modified $\boldsymbol{G}_N$-coset codes that adapts decoding effort to channel conditions and does not require an outer code. By storing and exploring flipping candidates in a min-heap and leveraging PM-based scores, SCOS closely approaches ML performance for code lengths up to $N\in\{64,128,256,512\}$ with average complexity near that of SC decoding; it also outperforms SC-Fano, SCS, and SCL in various regimes. The authors extend SCOS with a threshold-based maximum path metric to impose complexity constraints and improve error detection without an outer code, showing gains for PAC and dRM-polar codes over CRC-aided polar decoders at high SNR. Comparative results against RM and RM-polar codes demonstrate near-ML performance and substantial complexity reductions, with notable improvements when dynamic frozen bits are used. Overall, SCOS provides a practical, complexity-adaptive ML decoding framework for high-performance short-to-moderate length polar-like codes.

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.

Figures (9)

• Figure 1: (a) Initial SC decoding outputs $v^N = (0111)$ with the corresponding PM $M(v^N)$. (b) During the initial SC decoding, the PM and scores are computed for branch nodes $\left\{\textcolor{red}{1},0\textcolor{red}{0},01\textcolor{red}{0},011\textcolor{red}{0}\right\}$. (c) The branch nodes with PM larger than that of the current most likely leaf are pruned, e.g., we have $M(0\textcolor{red}{0}), M(011\textcolor{red}{0}) > M(0111)$. Suppose also that $S(01\textcolor{red}{0})<S(\textcolor{red}{1})$. Then, $\mathcal{L}$ stores all branch nodes with PM smaller than that of current most likely leaf, where $\mathcal{L}$ is a min heap according to the scores of its members. (d) The candidate with smallest score is popped from the heap and the decoder returns to the deepest (or nearest) common node. (e) The decision is flipped and SC decoding continues. During decoding, the heap$\mathcal{L}$ and the current most likely leaf are updated. (f) The branch nodes with PM larger than that of the current most likely leaf are pruned as in (c) (in this case, a leaf node is removed). (g) Repeat the procedure as in step (d), where we assume $M\left({1}\textcolor{blue}{1}\right)>M\left(0100\right)$. (h) The heap $\mathcal{L}$ is updated when the branch $\left({1}\textcolor{blue}{1}\right)$ was visited. (i) The decoder examines the last member of the heap $\mathcal{L}$ and pops it from $\mathcal{L}$. After reaching the $N$-th decoding phase, suppose that there is no branch node left, which has a smaller PM than that of the current most likely leaf, i.e., $\mathcal{L}=\varnothing$. The current most likely leaf is declared as the decision $\hat{u}^N$.
• Figure 2: FER/ANV/histogram for node visits vs. $E_b/N_0$ over the biAWGN channel for the $\left(128,64\right)$PAC code under SCOS decoding compared to relative RCU bound/lower bound \ref{['eq:complexity_average']}.
• Figure 3: FER/ANV vs. $E_b/N_0$ over the biAWGN channel for short dRM codes under SCOS decoding with $\lambda_\text{max}\textcolor{black}{\,=\eta }=10$ and $\lambda_\text{max}\textcolor{black}{\,=\eta}=100$ for $N=64$ and $N=128$, respectively, compared to relative RM codes under ML decoding and RCU bounds.
• Figure 4: FER/ANV vs. $E_b/N_0$ over the biAWGN channel for moderate-length dRM and dRM-polar codes under SCOS decoding $\lambda_\text{max}\textcolor{black}{\,=\eta}=5000$ compared to relative RM and RM-polar codes and RCU bounds.
• Figure 5: FER/ANV vs. $E_b/N_0$ over the biAWGN channel for the$\left(128,64\right)$PAC code under SCOS decoding with various bias terms and maximum complexity constraints such that $\eta = \lambda_{\text{max}}$.

Theorems & Definitions (6)

• Definition 1
• Remark 1
• Definition 2
• Lemma 1
• proof
• Remark 2