Deepcode (H. Kim et al. 2018) is a recently suggested Deep Learning-based scheme for communication over the AWGN channel with noisy feedback, claimed to be superior to all previous schemes in the literature. Deepcode's use of nonlinear coding (via Deep Learning) has been inspired by known shortcomings (Y.-H. Kim et al. 2007) of linear feedback schemes. In 2014, we presented a nonlinear feedback coding scheme based on a combination of the classical Schalwijk-Kailath (SK) scheme and modulo-arithmetic . This Modulo-SK scheme, which uses a small number of elementary operations and does employ any type of neural network, has been omitted from the performance comparisons made in the Deepcode paper, due to its use of common randomness (dither). However, the dither in the Modulo-SK scheme was used only for the standard purpose of tractable performance analysis, and is not required in practice. In this short note, we show that a fully-deterministic Modulo-SK (without any dithering) can outperform Deepcode. For example, to attain an error probability of 10^(-4) at rate 1/3 and feedforward SNR of 0dB, Modulo-SK requires 3dB less feedback SNR than Deepcode. To attain an error probability of 10^(-6) in the same setup but with noiseless feedback, Deepcode requires 150 rounds of communication, whereas Modulo-SK requires only 15, even if the feedback is noisy (with 27dB SNR). We further address the numerical stability issues of the original SK scheme reported in the Deepcode paper, and explain how they can be avoided. We augment this report with an online-available, fully-functional Matlab simulation for both the classical and Modulo-SK schemes. Finally, note that Modulo-SK is by no means claimed to be the best possible solution; in particular, using deep learning in conjunction with modulo-arithmetic might lead to better designs, and remains a fascinating direction for future research.

本文介绍了深度学习应用于 AWGN 信道上的通信方案 Deepcode，并提出了基于模运算和 SK 算法的非线性反馈编解码方案 Modulo-SK，得出结果表明后者优于前者，特别是在无噪声反馈情况下。同时，本文附带了深度学习与 Modulo-SK 的仿真代码用于进一步研究。

简明模数可以显著优于基于深度学习的Deepcode