Log-logarithmic Time Pruned Polar Coding on Binary Erasure Channels

TL;DR

This paper introduces a pruned polar coding scheme for binary erasure channels that achieves near-capacity performance with extremely low encoding and decoding complexity, especially for small error probabilities.

Contribution

It presents a novel pruned polar coding method that attains logarithmic time complexity in the inverse error probability for binary erasure channels.

Findings

Codes with block length $ ext{O}( ext{epsilon}^{-5})$

Error probability $ ext{O}( ext{epsilon})$

Encoding/decoding time $O( ext{log}| ext{log} ext{epsilon}|)$

Abstract

A pruned variant of polar coding is reinvented for all binary erasure channels. For small $\varepsilon>0$, we construct codes with block length $\varepsilon^{-5}$, code rate $\text{Capacity}-\varepsilon$, error probability $\varepsilon$, and encoding and decoding time complexity $O(N\log|\log\varepsilon|)$ per block, equivalently $O(\log|\log\varepsilon|)$ per information bit (Propositions 5 to 8). This result also follows if one applies systematic polar coding [Ar{\i}kan 10.1109/LCOMM.2011.061611.110862] with simplified successive cancelation decoding [Alamdar-Yazdi-Kschischang 10.1109/LCOMM.2011.101811.111480], and then analyzes the performance using [Guruswami-Xia arXiv:1304.4321] or [Mondelli-Hassani-Urbanke arXiv:1501.02444].