Constant Weight Polar Codes through Periodic Markov Processes

TL;DR

This paper investigates the polarization properties of constant weight codes generated from periodic Markov processes, demonstrating that fixing the initial state of the chain ensures polarization, unlike the general case.

Contribution

It establishes that fixing the initial state of a periodic Markov chain leads to polarization in constant weight codes, addressing a gap in previous understanding.

Findings

Polarization occurs when the initial state of the Markov chain is fixed.

General periodic Markov chains do not exhibit polarization.

Fixing the initial state aligns with constant weight code requirements.

Abstract

Constant weight codes can arise from an input process sampled from a periodic Markov chain. A previous result showed that, in general, polarization does not occur for input-output processes with an underlying periodic Markov chain. In this work, we show that if we fix the initial state of an underlying periodic Markov chain, polarization does occur. Fixing the initial state is aligned with ensuring a constant weight code.