Erasure Schemes Using Generalized Polar Codes: Zero-Undetected-Error Capacity and Performance Trade-offs

TL;DR

This paper analyzes the zero-undetected-error capacity of generalized polar codes for erasure channels, deriving formulas and trade-offs between error, erasure, and code rate.

Contribution

It provides a closed-form expression for the zero-undetected-error capacity of GP codes and characterizes the trade-offs between error and erasure probabilities.

Findings

Zero-undetected-error capacity formula for GP codes

Existence of codes with zero undetected error and low erasure probability below capacity

Trade-off between error probability and erasure probability near capacity

Abstract

We study the performance of generalized polar (GP) codes when they are used for coding schemes involving erasure. GP codes are a family of codes which contains, among others, the standard polar codes of Ar{\i}kan and Reed-Muller codes. We derive a closed formula for the zero-undetected-error capacity $I_0^{GP}(W)$ of GP codes for a given binary memoryless symmetric (BMS) channel $W$ under the low complexity successive cancellation decoder with erasure. We show that for every $R<I_0^{GP}(W)$, there exists a generalized polar code of blocklength $N$ and of rate at least $R$ where the undetected-error probability is zero and the erasure probability is less than $2^{-N^{\frac{1}{2}-\epsilon}}$. On the other hand, for any GP code of rate $I_0^{GP}(W)<R<I(W)$ and blocklength $N$, the undetected error probability cannot be made less than $2^{-N^{\frac{1}{2}+\epsilon}}$ unless the erasure probability is close to $1$.