Polar Codes For Broadcast Channels

TL;DR

This paper introduces polar codes tailored for broadcast channels, achieving capacity-bound rates with efficient encoding/decoding and low error probabilities, extending polar coding theory to multi-user communication scenarios.

Contribution

It develops polar coding schemes for broadcast channels, including deterministic and noisy cases, with capacity-achieving performance and practical complexity considerations.

Findings

Achieves rates on the boundary of the private-message capacity region.

Provides polar code implementations for superposition and Marton's schemes.

Ensures low error probability with stretched-exponential decay.

Abstract

Polar codes are introduced for discrete memoryless broadcast channels. For $m$-user deterministic broadcast channels, polarization is applied to map uniformly random message bits from $m$ independent messages to one codeword while satisfying broadcast constraints. The polarization-based codes achieve rates on the boundary of the private-message capacity region. For two-user noisy broadcast channels, polar implementations are presented for two information-theoretic schemes: i) Cover's superposition codes; ii) Marton's codes. Due to the structure of polarization, constraints on the auxiliary and channel-input distributions are identified to ensure proper alignment of polarization indices in the multi-user setting. The codes achieve rates on the capacity boundary of a few classes of broadcast channels (e.g., binary-input stochastically degraded). The complexity of encoding and decoding is $O(n*log n)$ where $n$ is the block length. In addition, polar code sequences obtain a stretched-exponential decay of $O(2^{-n^{\beta}})$ of the average block error probability where $0 < \beta < 0.5$.