[Paper Review] Simple Low-Rate Non-Binary LDPC Coding for Relay Channels

Puripong Suthisopapan, Kenta Kasai+3 more|arXiv (Cornell University)|Aug 16, 2011

Cooperative Communication and Network Coding30 references3 citations

TL;DR

This paper proposes a simple low-rate non-binary LDPC (NBLDPC) coding strategy for decode-and-forward half-duplex relay channels using multiplicative repetition and regular (2, dc)-structured NBLDPC codes over GF(2⁸). The scheme achieves performance within 1.2 dB of the relay channel’s achievable rate and provides a 1.1 dB coding gain over direct transmission, with low-complexity decoding at both relay and destination and compatibility with rate-compatible HARQ via puncturing.

ABSTRACT

Binary LDPC coded relay systems have been well studied previously with the assumption of infinite codeword length. In this paper, we deal with non-binary LDPC codes which can outperform their binary counterpart especially for practical codeword length. We utilize non-binary LDPC codes and recently invented non-binary coding techniques known as multiplicative repetition to design the low-rate coding strategy for the decode-and-forward half-duplex relay channel. We claim that the proposed strategy is simple since the destination and the relay can decode with almost the same computational complexity by sharing the same structure of decoder. Numerical experiments are carried out to show that the performances obtained by non-binary LDPC coded relay systems surpass the capacity of direct transmission and also approach within less than 1.5 dB from the achievable rate of the relay channels.

Motivation & Objective

• Address the limitations of binary LDPC codes in practical relay systems, particularly their need for complex degree profile optimization and poor performance at short-to-moderate codeword lengths.
• Overcome the high latency and hardware complexity of optimized binary LDPC codes by leveraging non-binary LDPC codes with regular structure.
• Design a low-rate coding strategy that enables efficient, low-complexity decoding at both relay and destination using shared Tanner graph structures.
• Achieve near-capacity performance with minimal design overhead and without requiring outer BCH or RS codes to suppress error floors.
• Enable rate-compatible HARQ operation through puncturing and adjustable repetition parameters for dynamic channel adaptation.

Proposed method

• Utilizes (2, dc)-regular non-binary LDPC codes over GF(2⁸) with constant column weight 2, which are empirically known to perform well at short-to-moderate lengths.
• Employs multiplicative repetition at the relay: the relay re-encodes and forwards the decoded codeword by repeating it T times, increasing the effective rate to Rr = R_S,BC / T.
• Applies puncturing to the (2, dc)-regular mother code of rate R = 1/3 to generate lower-rate codes (e.g., R = 1/4) for rate-compatible operation.
• Uses the same Tanner graph structure for decoding at both the relay and the destination, ensuring nearly identical computational complexity.
• Employs density evolution and Monte Carlo simulations to evaluate performance, focusing on BER and FER across various code rates and SNR levels.
• Validates the strategy using numerical experiments with codeword lengths of k = 1024 bits and repetition factors T = 2, 3, etc.

Experimental results

Research questions

• RQ1 Can non-binary LDPC codes with regular structure outperform optimized binary LDPC codes in low-rate relay channels at practical codeword lengths?
• RQ2 To what extent can multiplicative repetition and puncturing enable rate-compatible HARQ in decode-and-forward relay systems without sacrificing performance?
• RQ3 Can (2, dc)-regular NBLDPC codes achieve near-capacity performance without degree profile optimization or outer codes?
• RQ4 How does the computational complexity of decoding compare between the relay and destination when using the same code structure?
• RQ5 What is the performance gap between the proposed scheme and the theoretical achievable rate of the relay channel at moderate codeword lengths?

Key findings

• The proposed NBLDPC-coded relay system achieves a bit error rate (BER) performance within 1.2 dB of the relay channel’s achievable rate at a BER of 10⁻⁵ with an information length of k = 1024 bits and rate Rr = 1/6.
• The system achieves a coding gain of approximately 1.1 dB over the Shannon capacity of direct transmission, demonstrating significant reliability improvement.
• The frame error rate (FER) performance is excellent across both the waterfall and error floor regions without the need for outer BCH or RS codes.
• Punctured codes for rate Rr = 1/4 maintain a coding gain of about 0.75 dB beyond direct transmission, confirming the effectiveness of rate-compatible design.
• The threshold for Rr = 1/6 is only 0.3 dB below the achievable rate, indicating strong convergence and near-optimal performance under density evolution analysis.
• The relay and destination achieve nearly identical decoding complexity due to shared Tanner graph structure, enabling efficient and scalable implementation.