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 SK
scheme and modulo-arithmetic, using a small number of elementary operations
without any type of neural network. This Modulo-SK scheme has been omitted from
the performance comparisons made in the Deepcode paper, due to its use of
common randomness (dither), and in a later version since it was incorrectly
interpreted as a variable-length coding scheme. However, the dither in
Modulo-SK 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 dithering) can outperform Deepcode. For
example, to attain an error probability of 10^(-4) at rate 1/3 Modulo-SK
requires 3dB less feedback SNR than Deepcode. To attain an error probability of
10^(-6) with noiseless feedback, Deepcode requires 150 rounds of communication,
whereas Modulo-SK requires only 15 rounds, 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

Simple Modulo can Significantly Outperform Deep Learning-based Deepcode

We present a comparative study of the performance of various polar code
constructions in an additive white Gaussian noise (AWGN) channel. A polar code
construction is any algorithm that selects $K$ best among $N$ possible polar
bit-channels at the design signal-to-noise-ratio (design-SNR) in terms of bit
error rate (BER). Optimal polar code construction is hard and therefore many
suboptimal polar code constructions have been proposed at different
computational complexities. Polar codes are also non-universal meaning the code
changes significantly with the design-SNR. However, it is not known which
construction algorithm at what design-SNR constructs the best polar codes. We
first present a comprehensive survey of all the well-known polar code
constructions along with their full implementations. We then propose a
heuristic algorithm to find the best design-SNR for constructing best possible
polar codes from a given construction algorithm. The proposed algorithm
involves a search among several possible design-SNRs. We finally use our
algorithm to perform a comparison of different construction algorithms using
extensive simulations. We find that all polar code construction algorithms
generate equally good polar codes in an AWGN channel, if the design-SNR is
optimized.

本文系统比较了不同设计信噪比条件下各种极化码构造的性能，并提出了一种启发式算法来选取最佳的设计信噪比条件以构造极化码，并进行了广泛的模拟比较，发现在信噪比优化的情况下，所有的极化码构造算法在加性白高斯噪声信道中生成的极化码的性能都相当好。

极化码在 AWGN 信道下构造方式的比较研究

A Comparative Study of Polar Code Constructions for the AWGN Channel

The use of open-loop coding can be easily extended to a closed-loop
concatenated code if the channel has access to feedback. This can be done by
introducing a feedback transmission scheme as an inner code. In this paper,
this process is investigated for the case when a linear feedback scheme is
implemented as an inner code and, in particular, over an additive white
Gaussian noise (AWGN) channel with noisy feedback. To begin, we look to derive
the optimal linear feedback scheme by optimizing over the received
signal-to-noise ratio. From this optimization, an asymptotically optimal linear
feedback scheme is produced and compared to other well-known schemes. Then, the
linear feedback scheme is implemented as an inner code to a concatenated code
over the AWGN channel with noisy feedback. This code shows improvements not
only in error exponent bounds, but also in bit-error-rate and frame-error-rate.
It is also shown that the if the concatenated code has total blocklength L and
the inner code has blocklength, N, the inner code blocklength should scale as N
= O(C/R), where C is the capacity of the channel and R is the rate of the outer
code. Simulations with low density parity check (LDPC) and turbo codes are
provided to display these advantages.

本文研究了在加性白噪声信道上实现线性反馈编码作为内嵌编码的最佳方法，并比较了其他著名方案的性能。结果表明，该方法不仅在误码率和帧错误率上有所改善，而且在误差指数方面也有所提高。此外，模拟结果证明内嵌编码块长度应当按照 N=O (C/R) 的比例进行扩展，其中 C 为信道容量，R 为外部编码速率。