# Hsin-Po Wang Math PhD @ UIUC

From the newest to the oldest.

Abbreviation Title
Hypotenuse19 Polar Codes’ Simplicity, Random Codes’ Durability
LoglogTime19 Log-logarithmic Time Pruned Polar Coding
LargeDevia18 Polar-like Codes and Asymptotic Tradeoff among Block Length, Code Rate, and Error Probability
LoglogTime18 Log-logarithmic Time Pruned Polar Coding on Binary Erasure Channels
ModerDevia18 Polar Code Moderate Deviation: Recovering the Scaling Exponent

ModerDevia18 focuses on the moderate deviations regime (MDR) of polar coding. MDR is also called the moderate deviations principle (MDP) paradigm in some references. It discusses about the relation among block length ($N$), error probability ($P$), and code rate ($R$) in the region where $P$ is about $\exp(-N^\pi)$ and $R$ is about $\text{Capacity} - N^{-\rho}$ for some positive numbers $\pi,\rho$. The precise goal is to characterize the region of $(\pi,\rho)$ pairs that are achievable for $N\to\infty$.

While ModerDevia18 deals with the classical polar codes constructed in Arıkan’s original paper, LargeDevia18 extends the theory to a wide class of polar codes. We are able to predict, up to some big-$O$ notations, how codes constructed with a certain kernel $G$ will behave given the scaling exponent $\mu$ (or its inverse $\rho=1/\mu$) and the partial distances. It does not mean that such prediction is easy to make because finding the precise $\rho$ (or $\mu$) is difficult. That said, bounding $\rho$ is easy, so is bounding the MDP behavior.

LoglogTime18 stands on the result of ModerDevia18 and shows that, if we would like to tolerate higher $P$ and lower $R$, we can reduce the complexity of the encoding and decoding from $\log N$ to $\log(\log N)$, per information bit. By higher $P$ we mean $P$ scales as $N^{-1/5}$; By lower $R$ we mean $R$ scales as $\text{Capacity}-N^{-1/5}$. Thus the constructed codes still achieve capacity.

While LoglogTime18 deals with the binary erasure channels, LoglogTime19 handles arbitrary symmetric $p$-ary channels, where $p$ is any prime. The latter result is similar: by tolerating that $P$ converge to $0$ slower and that $R$ that converge to the capacity slower, we can reduce the complexity to $\log(\log N)$ per information bit. In both LoglogTime18 and LoglogTime19, codes are construct with the standard kernel $[^1_1{}^0_1]$.

We later found (not included in either paper) that the conclusion generalizes to arbitrary discrete memoryless channels.

Hypotenuse19 shows that it is possible to construct codes whose error probabilities and code rates scale like random codes’ and encoding and decoding complexities scale like polar codes’. On one hand, random codes’ error and rate are considered the optimal. On the other, polar codes’ complexity ($\log N$) is considered low. (Not the lowest possible complexity, as there exist $\log(\log N)$ constructions for general channels and $O(1)$ constructions for BEC.) This result holds for all discrete memoryless channels, the family of channels Shannon considered in 1948. This result extends a series of works done by (alphabetically) Arıkan, Błasiok, Fazeli, Guruswami, Hassani, Honda, Korada, Mori, Şaşoğlu, Sutter, etc.

For a quick comparison, see Figure 1 on page 3 in Hypotenuse19. See also Table 2 on page 40. 