# Log-logarithmic Time Pruned Polar Coding

## TL;DR

This paper introduces a pruned polar coding method for binary erasure channels that achieves capacity with extremely low encoding and decoding complexity, specifically log-logarithmic in key parameters, marking a significant advancement.

## Contribution

It presents the first explicit family of capacity-achieving codes with log-logarithmic per-bit complexity, extending the concept to q-ary symmetric channels and achieving the fastest known complexity.

## Key findings

- Capacity-achieving codes with block length N=ε^{-5} for small ε
- Per-bit complexity is O(log|log ε|), the second lowest after repeat-accumulate codes
- Generalization to q-ary channels with complexity O(log|log ε|)

## Abstract

A pruned variant of polar coding is proposed for binary erasure channels. For sufficiently small $\varepsilon>0$, we construct a series of capacity achieving codes with block length $N=\varepsilon^{-5}$, code rate $R=\text{Capacity}-\varepsilon$, error probability $P=\varepsilon$, and encoding and decoding time complexity $\text{bC}=O(\log\left|\log\varepsilon\right|)$ per information bit. The given per-bit complexity $\text{bC}$ is log-logarithmic in $N$, in $\text{Capacity}-R$, and in $P$; no known family of codes possesses this property. It is also the second lowest $\text{bC}$ after repeat-accumulate codes and their variants. While random codes and classical polar codes are the only two families of capacity-achieving codes whose $N$, $R$, $P$, and $\text{bC}$ were written down as explicit functions, our construction gives the third family. Then we generalize the result to: Fix a prime $q$ and fix a $q$-ary-input discrete symmetric memoryless channel. For sufficiently small $\varepsilon>0$, we construct a series of capacity achieving codes with block length $N=\varepsilon^{-O(1)}$, code rate $R=\text{Capacity}-\varepsilon$, error probability $P=\varepsilon$, and encoding and decoding time complexity $\text{bC}=O(\log\left|\log\varepsilon\right|)$ per information bit. The later construction gives the fastest family of capacity-achieving codes to date on those channels.

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Source: https://tomesphere.com/paper/1905.13340