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Constant Weight Polar Codes through Periodic Markov Processes

Boaz Shuval, Ido Tal

TL;DR

This work shows that polarization can occur for input processes driven by periodic Markov chains if the initial state is fixed, enabling constant-weight constrained codes. It extends Arikan-style polarization to periodic-state models by decomposing the period and applying slow and fast polarization analyses conditioned on the initial phase, leveraging the Honda–Yamamoto scheme to achieve rates near $\mathcal{H}_{X} - \mathcal{H}_{X|Y}$. The key contributions are (i) a phase-conditioned slow polarization yielding entropy-rate fixed points and (ii) a phase-conditioned fast polarization establishing rapid convergence of $Z$ and $K$ metrics, together with a reduction argument showing conditioning on the initial state preserves the overall entropy rate. The results broaden the applicability of capacity-achieving polar codes to constant-weight and other constrained sequence scenarios in finite-state, periodic settings, with encoding complexity $O(|\mathcal{S}|^3 N \log N)$.

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.

Constant Weight Polar Codes through Periodic Markov Processes

TL;DR

This work shows that polarization can occur for input processes driven by periodic Markov chains if the initial state is fixed, enabling constant-weight constrained codes. It extends Arikan-style polarization to periodic-state models by decomposing the period and applying slow and fast polarization analyses conditioned on the initial phase, leveraging the Honda–Yamamoto scheme to achieve rates near . The key contributions are (i) a phase-conditioned slow polarization yielding entropy-rate fixed points and (ii) a phase-conditioned fast polarization establishing rapid convergence of and metrics, together with a reduction argument showing conditioning on the initial state preserves the overall entropy rate. The results broaden the applicability of capacity-achieving polar codes to constant-weight and other constrained sequence scenarios in finite-state, periodic settings, with encoding complexity .

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.
Paper Structure (6 sections, 15 theorems, 79 equations, 3 figures)

This paper contains 6 sections, 15 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and Main Theorem
  3. Proof of Slow Polarization
  4. Proof of Fast Polarization
  5. Proof of \ref{['thm:main']}
  6. Auxiliary Proofs

Key Result

Theorem 1

Let $(X_j, Y_j, S_j)_{j \in \mathbb{Z}}$ be a FIM process, where $(S_j)_{j \in \mathbb{Z}}$ has period $\mathsf{p}$. For $N = 2^n$, denote by $U_1^N$ the polar transform of $X_1^N$, and fix $0 < \beta <1/2$. Fix $0 \leq \varphi < \mathsf{p}$, and a non-empty set $\Psi_0 \subseteq \mathcal{S}$ such t Then,

Figures (3)

  • Figure 1: Markov chain for producing sequences of weight $N/2$. Labels for blue and red arrows are '$0$' and '$1$', respectively. Transition probabilities between states are shown on the arrows. The state $\varepsilon$ is the start and end state, at which the output is ensured to have weight $N/2$. The chain has period $\mathsf{p}=4$; each horizontal layer of the graph is a different phase $\phi$.
  • Figure 2: Condensed Markov chain for producing sequences of weight $N/2$. Here, states from \ref{['fig: weight half Markov chain']} of the same phase and weight are merged. The probability of a blue (red) edge from a node with phase $\phi$ and weight $w$ to a node with phase $\phi+1$ and weight $w$ ($w+1$) is the number of valid sequences, i.e. length $4$ and weight $2$, whose weight in the prefix of length $\phi+1$ is $w$ ($w+1$) divided by the number of valid sequences whose weight in the prefix of length $\phi$ is $w$.
  • Figure 3: Markov chain for producing sequences whose weight modulo $b$ is constrained to some integer $0 \leq a < b$. Blue and red arrows are labelled '$0$' and '$1$', respectively. All edge probabilities are $1/2$. The chain is aperiodic. To construct the sequences, we constrain the initial state to '$0$' and the final state to '$a$'.

Theorems & Definitions (16)

  • Theorem 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Corollary 7
  • Lemma 8
  • Lemma 9
  • ...and 6 more