Modular Arithmetic Erasure Channels and Their Multilevel Channel Polarization

TL;DR

This paper introduces modular arithmetic erasure channels (MAECs), generalizing binary erasure channels, and demonstrates their multilevel polarization behavior, providing recursive formulas and solving an open problem in non-binary polar codes.

Contribution

It proposes MAECs, derives recursive polar transform formulas for them, and establishes a method to analyze multilevel channel polarization, addressing an open problem in non-binary polar coding.

Findings

MAECs generalize binary erasure channels.

Recursive formulas for polar transforms of MAECs are derived.

Method to compute limiting proportions of polarized channels is established.

Abstract

This study proposes \emph{modular arithmetic erasure channels} (MAECs), a novel class of erasure-like channels with an input alphabet that need not be binary. This class contains the binary erasure channel (BEC) and some other known erasure-like channels as special cases. For MAECs, we provide recursive formulas of Ar{\i}kan-like polar transform to simulate channel polarization. In other words, we show that the synthetic channels of MAECs are equivalent to other MAECs. This is a generalization of well-known recursive formulas of the polar transform for BECs. Using our recursive formulas, we also show that a recursive application of the polar transform for MAECs results in \emph{multilevel channel polarization,} which is an asymptotic phenomenon that is characteristic of non-binary polar codes. Specifically, we establish a method to calculate the limiting proportions of the partially noiseless and noisy channels that are generated as a result of multilevel channel polarization for MAECs. In the particular case of MAECs, this calculation method solves an open problem posed by Nasser (2017) in the study of non-binary polar codes.