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Successive Cancellation Sampling Decoder: An Attempt to Analyze List Decoding Theoretically

Hsin-Po Wang, Venkatesan Guruswami

TL;DR

This work tackles the difficulty of theoretically analyzing successive cancellation list decoders for polar codes by introducing successive cancellation sampling (SCS), a parallel, agent-based decoder that samples codewords from posterior probabilities. The authors derive an explicit upper bound on the SCS error probability gap relative to an optimal $\ell$-list decoder and show that temperature-based adjustments (SCIS) can further reduce errors. They analyze SCS under natural posterior mass distributions (geometric and zeta) and study the asymptotic behavior as the number of agents grows, providing insight into when and how the method approaches the performance of larger lists. The proposed framework offers a tractable bridge between practical decoder designs and theoretical analysis, with potential extensions to PAC codes and other successively decoded schemes.

Abstract

Successive cancellation list (SCL) decoders of polar codes excel in practical performance but pose challenges for theoretical analysis. Existing works either limit their scope to erasure channels or address general channels without taking advantage of soft information. In this paper, we propose the "successive cancellation sampling" (SCS) decoder. SCS hires iid "agents" to sample codewords using posterior probabilities. This makes it fully parallel and amenable for some theoretical analysis. As an example, when comparing SCS with $a$ agents to any list decoder with list size $\ell$, we can prove that the error probability of the former is at most $\ell/ae$ more than that of the latter. In this paper, we also describe how to adjust the "temperature" of agents. Warmer agents are less likely to sample the same codewords and hence can further reduce error probability.

Successive Cancellation Sampling Decoder: An Attempt to Analyze List Decoding Theoretically

TL;DR

This work tackles the difficulty of theoretically analyzing successive cancellation list decoders for polar codes by introducing successive cancellation sampling (SCS), a parallel, agent-based decoder that samples codewords from posterior probabilities. The authors derive an explicit upper bound on the SCS error probability gap relative to an optimal -list decoder and show that temperature-based adjustments (SCIS) can further reduce errors. They analyze SCS under natural posterior mass distributions (geometric and zeta) and study the asymptotic behavior as the number of agents grows, providing insight into when and how the method approaches the performance of larger lists. The proposed framework offers a tractable bridge between practical decoder designs and theoretical analysis, with potential extensions to PAC codes and other successively decoded schemes.

Abstract

Successive cancellation list (SCL) decoders of polar codes excel in practical performance but pose challenges for theoretical analysis. Existing works either limit their scope to erasure channels or address general channels without taking advantage of soft information. In this paper, we propose the "successive cancellation sampling" (SCS) decoder. SCS hires iid "agents" to sample codewords using posterior probabilities. This makes it fully parallel and amenable for some theoretical analysis. As an example, when comparing SCS with agents to any list decoder with list size , we can prove that the error probability of the former is at most more than that of the latter. In this paper, we also describe how to adjust the "temperature" of agents. Warmer agents are less likely to sample the same codewords and hence can further reduce error probability.
Paper Structure (18 sections, 5 theorems, 31 equations, 4 figures)

This paper contains 18 sections, 5 theorems, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Walking on the Tree
  3. Polar codes and SC
  4. SCL decoder: more chances to win
  5. SCS, a new decoder
  6. Agents restart a lot, or do they?
  7. How Good is SCS?
  8. Proof of Theorem \ref{['thm:optimal']}
  9. Is Theorem \ref{['thm:optimal']} tight?
  10. SCS and Natural PMFs
  11. Geometric distributions
  12. Zeta distributions
  13. Limiting behavior when there are many agents
  14. Customize the Temperature
  15. Twist the distribution
  16. ...and 3 more sections

Key Result

Theorem 1

We say that a list decoder errsWe are using the definition of given by Elias in Eli91. if the correct message is not in the final list. Let $\ell$ and $a$ be positive integers. Comparing the SCS decoder with $a$ agents to any list decoderWithout specifying SC, we mean all possible list decoders, inc more than that of the latter.

Figures (4)

  • Figure 1: The tree of posterior probabilities. Msg stands for message. The usual SC decoder chooses the path greedily $51\% \to 38\% \to 82\% \to 76\% \to 64\%$. The SCS decoder, however, chooses this path with probability $51\% \cdot 38\% \cdot 82\% \cdot 76\% \cdot 64\%$.
  • Figure 2: Left: Agent must restart. Right: Agent might default to the compatible branch.
  • Figure 3: The functions $z (1 - z)^a$ and a concave increasing upper bound $g(z)$.
  • Figure 4: Error probability as a function of $\beta$. We use geometric distribution with success rate $1 - q = 0.1$. One curve for a fixed $a$. From top to bottom: $a = 1, 2, 4, \dotsc, 256$.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • Proposition 3
  • Proposition 4
  • Lemma 5