A Universal List Decoding Algorithm with Application to Decoding of Polar Codes

TL;DR

The paper addresses the challenge of decoding short linear block codes efficiently across a range of code rates, introducing Guessing Codeword Decoding (GCD) as an ML-inspired, universal list-decoding approach that enumerates lightest test error patterns via partial re-encodings. It develops an ordered TEP generator based on a Flipping Pattern Tree, analyzes complexity, and proposes truncation (ell_max, tau_s, tau_p) and parallelization to reduce queries and latency, with theoretical bounds derived using saddlepoint approximations. The authors extend GCD to polar codes through pruning and a multiple-bit-wise SCL decoding scheme, achieving comparable performance to CA-SCL while significantly reducing decoding latency, including parallel processing and early stopping criteria. The empirical results across RM and polar codes show no performance loss relative to ML decoding in many regimes and substantial latency reductions, highlighting the approach’s practicality for URLLC and long codes built from short blocks.

Abstract

This paper is concerned with a guessing codeword decoding (GCD) of linear block codes. Compared with the guessing noise decoding (GND), which is only efficient for high-rate codes, the GCD is efficient for not only high-rate codes but also low-rate codes. We prove that the GCD typically requires a fewer number of queries than the GND. Compared with the ordered statistics decoding (OSD), the GCD does not require the online Gaussian elimination (GE). In addition to limiting the maximum number of searches, we suggest limiting the radius of searches in terms of soft weights or tolerated performance loss to further reduce the decoding complexity, resulting in the so-called truncated GCD. The performance gap between the truncated GCD and the optimal decoding can be upper bounded approximately by the saddlepoint approach or other numerical approaches. The derived upper bound captures the relationship between the performance and the decoding parameters, enabling us to balance the performance and the complexity by optimizing the decoding parameters of the truncated GCD. We also introduce a parallel implementation of the (truncated) GCD algorithm to reduce decoding latency without compromising performance. Another contribution of this paper is the application of the GCD to the polar codes. We propose a multiple-bit-wise decoding algorithm over a pruned tree for the polar codes, referred to as the successive-cancellation list (SCL) decoding algorithm by GCD. First, we present a strategy for pruning the conventional polar decoding tree based on the complexity analysis rather than the specific bit patterns. Then we apply the GCD algorithm in parallel aided by the early stopping criteria to the leaves of the pruned tree. Simulation results show that, without any performance loss as justified by analysis, the proposed decoding algorithm can significantly reduce the decoding latency of the polar codes.

Figures (24)

• Figure 1: The update process of the linked list $\mathcal{L}$.
• Figure 2: Average number of queries for the GND (SGRAND) and the GCD of the RM codes of length $N = 32$ and dimension $K \in \{6, 16, 26\}$. Here, the channel is BPSK-AWGN, the target $\text{FER}=10^{-3}$ and $L =1$.
• Figure 3: A rooted tree with $K=4$.
• Figure 4: An example of the FPT algorithm to generate an ordered list of the five best TEPs $\{0000,1000,0100,0010,1100\}$. Here, $\mathcal{M}$ is a linked list and $\mathcal{T}$ is an ordered rooted tree.
• Figure 5: ML decoding of the RM codes $\mathscr{C}_{\rm{RM}}[64,K]$ with OSD and GCD ($L=1$).

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Example 1
• Example 2
• Example 3
• Example 4
• Lemma 1
• Example 5
• Example 6
• Example 7