Rate-Matching Deep Polar Codes via Polar Coded Extension

TL;DR

This work tackles the length constraint of deep polar codes by introducing a rate-matching extension that concatenates multi-layer pre-transformed codewords to achieve non-power-of-two lengths while preserving deep polarization benefits. It combines an LLR-enhanced SoSCL decoding framework with a DEGA-based design methodology to optimize the rate profile across layers, supported by a greedy algorithm that reduces design complexity from exponential to linear in the number of layers. Theoretical analysis via density evolution under Gaussian approximation and extensive simulations demonstrate substantial coding gains over conventional rate-matching methods, particularly at medium to high code rates and short blocklengths. The approach enables flexible, high-performance coding suitable for URLLC and short-packet communications with manageable encoding/decoding complexity.

Abstract

Deep polar codes are pre-transformed polar codes that employ a multi-layered polar kernel transformation strategy to enhance code performance in short blocklength regimes. However, like conventional polar codes, their block length is constrained to powers of two, as the final transformation layer uses a conventional polar kernel matrix. This paper introduces a novel rate-matching technique for deep polar codes using code extension, particularly effective when the desired code length slightly exceeds a power of two. The key idea is to exploit the layered structure of deep polar codes by concatenating polar codewords generated at each transformation layer. Based on this structure, we also develop an efficient decoding algorithm leveraging soft-output successive cancellation list decoding and provide comprehensive error probability analysis supporting our code design algorithms. Additionally, we propose a computationally efficient greedy algorithm for multi-layer configurations. Extensive simulations confirm that our approach delivers substantial coding gains over conventional rate-matching methods, especially in medium to high code-rate regimes.

Figures (11)

• Figure 1: Factor graph of polar transform and its corresponding binary tree, whose nodes consist of processing elements.
• Figure 2: Illustration of the encoding for single-layer extended deep polar codes.
• Figure 3: The proposed LLR combined decoding process.
• Figure 4: An example of extended deep polar codes with parameters $(N_0, N_1, K)=(8,4,3)$ under a BEC with an erasure probability of $\epsilon=0.5$.
• Figure 5: The encoding for multi-layer extended deep polar codes.

Theorems & Definitions (3)

• Remark 1: Effect of Soft Information $\Lambda_i$
• Remark 2
• Remark 3: Parameters of deep polar codes given $M$