Group Probability Decoding of Turbo Product Codes over Higher-Order Fields

TL;DR

This work addresses the loss of bit-level information in traditional turbo product decoding by introducing group-probability decoding, which preserves correlations in the codeword distribution. It provides a theoretical framework distinguishing exogenous and endogenous correlation, derives achievable information rates for different preprocessing schemes, and demonstrates that group probabilities can yield Eb/N0 gains in both correlation regimes. The authors develop non-binary turbo product codes and implement group-probability decoding using SISO decoders such as ORBGRAND-AI and SOGRAND, with a Group-Probability SISO decoder framework and Group-Probability SOGRAND decoding. Simulation results show gains up to $0.7$ dB for exogenous correlation and up to $0.3$ dB for endogenous correlation, validating the approach and highlighting its practical potential for low-rate, non-binary product codes in correlated channels.

Abstract

Binary turbo product codes (TPCs) are powerful error-correcting codes constructed from short component codes. Traditionally, turbo product decoding passes log likelihood ratios (LLRs) between the component decoders, inherently losing information when bit correlation exists. Such correlation can arise exogenously from sources like intersymbol interference and endogenously during component code decoding. To preserve these correlations and improve performance, we propose turbo product decoding based on group probabilities. We theoretically predict mutual information and signal-to-noise ratio (SNR) gains of group over bit-probability decoding. To translate these theoretical insights to practice, we revisit non-binary TPCs that naturally support group-probability decoding. We show that any component list decoder that takes group probabilities as input and outputs block-wise soft-output can partially preserve bit correlation, which we demonstrate with symbol-level ORBGRAND combined with soft-output GRAND (SOGRAND). Our results demonstrate that group-probability-based turbo product decoding achieves SNR gains of up to 0.3 dB for endogenous correlation and 0.7 dB for exogenous correlation, compared to bit-probability decoding.

Figures (12)

• Figure 1: Preprocessing schemes: Preprocessing (gw) calculates the APP of a group of modulation symbols $X_s^e$ conditioned on a window of received symbols $Y_s^e$. Preprocessing (bw) and (bs) treat individual modulation symbols as independent and can be used by a bit-probability decoder. As shown in Theorem \ref{['thm:information-rate']}, the mutual information between channel input and preprocessing output decreases from processing scheme (gw) to (bw) and from (bw) to (bs).
• Figure 2: Calculation of the achievable information rates $I_\text{gw}$, $I_\text{bw}$, and $I_\text{bs}$: Because $Y_i$ is strong stationary the GMI for $s=1$ is equal to the mutual information between one group or symbol of the channel input and output.
• Figure 3: Achievable information rate $I(E_\textnormal{b} / N_0)$ in bits per coded bit for each preprocessing scheme for a group size of $g=2$ modulation symbols and a Gauss-Markov channel with $\rho=0.75$. The vertical lines indicate the $E_\textnormal{b} / N_0$ threshold $(E_\textnormal{b} / N_0)^\star$ that is required for error-free communication of a long code of rate $r=0.9$. The SNR gain between scheme (gw) and (bw) ($t_\text{gw} - t_\text{bw}$) and between (bw) and (bs) ($t_\text{bw} - t_\text{bs}$) can be found in Fig. \ref{['fig:snr-gain']}.
• Figure 4: $E_\textnormal{b} / N_0$ gains between the preprocessing schemes for a group size $g=2$ symbols: The bars show the $E_\textnormal{b} / N_0$ savings for error-free communication of a long code of rate $r=0.9$. The black dashed lines show the actual gains for decoding an $(144, 130)$RLCode with ORBGRAND-AI.
• Figure 5: Information processing chain: information loss under group and bit marginalization: The decoder outputs a list of codewords and corresponding APP. The codeword APP are marginalized to group probabilities for consecutive groups of size $g$, followed by a marginalization to bit probabilities. According to the chain and the data-processing inequality SWB-1644451026, the mutual information between the decoder output and the probabilities can only remain constant or decrease from group to bit probabilities.

Theorems & Definitions (5)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• proof