# Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel

## TL;DR

This paper establishes that the known upper bound of 4.714 on the scaling exponent for binary-input memoryless channels also applies to the AWGN channel, and explores polar codes' performance in moderate deviations regimes.

## Contribution

It proves that the upper bound of 4.714 on the scaling exponent applies to the AWGN channel and introduces a construction balancing gap to capacity and error decay.

## Key findings

- The scaling exponent upper bound of 4.714 holds for the AWGN channel.
- Polar codes can achieve capacity within an $O(n^{-1/\mu}\sqrt{\log n})$ gap using $n$ constellations.
- A tradeoff construction for polar codes in the moderate deviations regime is proposed.

## Abstract

This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent $\mu$ of polar codes for a memoryless channel $q_{Y|X}$ with capacity $I(q_{Y|X})$ characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed $\varepsilon \in (0, 1)$, the gap between the capacity $I(q_{Y|X})$ and the maximum non-asymptotic rate $R_n^*$ achieved by a length-$n$ polar code with average error probability $\varepsilon$ scales as $n^{-1/\mu}$, i.e., $I(q_{Y|X})-R_n^* = \Theta(n^{-1/\mu})$. It is well known that the scaling exponent $\mu$ for any binary-input memoryless channel (BMC) with $I(q_{Y|X})\in(0,1)$ is bounded above by $4.714$, which was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the scaling exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of $O(n^{-1/\mu}\sqrt{\log n})$ by using an input alphabet consisting of $n$ constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of $n$ constellations can be achieved within a gap of $O(n^{-1/\mu}\log n)$ by using a superposition of $\log n$ binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as $n$ grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.