# Adaptive Learned Belief Propagation for Decoding Error-Correcting Codes
## Appendix C Parameters of Codes  
For low-rate codes, first, the degree distributions of the Tanner graph are determined using the extrinsic information transfer (EXIT) chart [100]. The EXIT chart produces accurate results if \(n\to\infty\)[100]. For high-rate codes, we apply the stochastic EXIT chart, which in the short block length regime yields better coding gains compared to the deterministic variant [101]. For instance, while the EXIT chart suggests that the degree distributions of \(\mathcal{C}_{6}\) are optimal near \(E_{b}/N_{0}=3\) dB, the stochastic EXIT chart in Figure 12 suggests \(E_{b}/N_{0}=3.5\) dB. Indeed, at \(E_{b}/N_{0}=3\) dB, the check-node extrinsic information \(I_{C}\) intersects with variable-node extrinsic information \(I_{V}\) for the deterministic EXIT chart. The exponent matrices are obtained using the PEG algorithm, which takes the optimized degree distribution polynomials. The matrices below are vectorized row-wise. They can be unvectorized considering their dimensions. \(\mathcal{C}_{2}\): \(\lambda=4\), \(\omega=8\), \(M=403\), \(\Lambda(x)=x^{3}\), \(\Upsilon(x)=x^{7}\), \(\mathrm{P}_{2}\) below  
* (11) [PERSON]; [PERSON]; [PERSON] Learning to decode linear codes using deep learning. In Proceedings of the 54 th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA, 27-30 September 2016; pp. 341-346. [CrossRef]
* (12) [PERSON]; [PERSON]; [PERSON]; [PERSON] On deep learning-based channel decoding. In Proceedings of the 2017 51 st Annual Conference on Information Sciences and Systems (CISS), Baltimore, MD, USA, 22-24 March 2017; pp. 1-6. [CrossRef]
* (13) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Communication algorithms via deep learning. In Proceedings of the International Conference on Learning Representations, Vancouver, BC, Canada, 30 April-3 May 2018. Available online: [[https://openreview.net/forum?id=ryazCMbR-](https://openreview.net/forum?id=ryazCMbR-)]([https://openreview.net/forum?id=ryazCMbR-](https://openreview.net/forum?id=ryazCMbR-)) (accessed on 11 June 2025). * (14) [PERSON]; [PERSON]; [PERSON] Learning to decode LDPC codes with finite-alphabet message passing. In Proceedings of the 2018 Information Theory and Applications Workshop (ITA), San Diego, CA, USA, 11-16 February 2018; pp. 1-9. * (15) [PERSON]; [PERSON]; [PERSON] Deep learning for decoding of linear codes--A syndrome-based approach. In Proceedings of the 2018 IEEE International Symposium on Information Theory (ISIT), Vail, CO, USA, 17-22 June 2018; pp. 1595-1599. [CrossRef]
* (16) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Deep learning methods for improved decoding of linear codes. _IEEE J. Sel. Top. Signal Process._**2018**, 12, 119-131. [CrossRef]
* (17) [PERSON] Learning Algorithms for Error Correction. Master's Thesis, McGill University, Montreal, QC, Canada, 2018. Available online: [[https://escholarship.mcgill.ca/concern/theses/c247](https://escholarship.mcgill.ca/concern/theses/c247) dv63d]([https://escholarship.mcgill.ca/concern/theses/c247](https://escholarship.mcgill.ca/concern/theses/c247) dv63d) (accessed on 11 June 2025). * (18) [PERSON]; [PERSON]; [PERSON]; [PERSON] Learned belief-propagation decoding with simple scaling and SNR adaptation. In Proceedings of the 2019 IEEE International Symposium on Information Theory (ISIT), Paris, France, 7-12 July 2019; pp. 161-165. [CrossRef]
* (19) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] DEEPTURBO: Deep Turbo decoder. In Proceedings of the 2019 IEEE 20 th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Cannes, France, 2-5 July 2019; pp. 1-5. [CrossRef]
* (20) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] Reinforcement learning for channel coding: Learned bit-flipping decoding. In Proceedings of the 2019 57 th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA, 24-27 September 2019; pp. 922-929.