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Stronger Polarization for the Deletion Channel

Dar Arava, Ido Tal

TL;DR

A polar coding scheme for the deletion channel with a probability of error that decays roughly like ${2^{ - \sqrt \lambda }}$ , where Λ is the length of the codeword, and is capacity-achieving.

Abstract

In this paper we show a polar coding scheme for the deletion channel with a probability of error that decays roughly like $2^{-\sqrtΛ}$, where $Λ$ is the length of the codeword. That is, the same decay rate as that of seminal polar codes for memoryless channels. This is stronger than prior art in which the square root is replaced by a cube root. Our coding scheme is similar yet distinct from prior art. The main differences are: 1) Guard-bands are placed in almost all polarization levels; 2) Trellis decoding is applied to the whole received word, and not to segments of it. As before, the scheme is capacity-achieving. The price we pay for this improvement is a higher decoding complexity, which is nonetheless still polynomial, $O(Λ^4)$.

Stronger Polarization for the Deletion Channel

TL;DR

A polar coding scheme for the deletion channel with a probability of error that decays roughly like , where Λ is the length of the codeword, and is capacity-achieving.

Abstract

In this paper we show a polar coding scheme for the deletion channel with a probability of error that decays roughly like , where is the length of the codeword. That is, the same decay rate as that of seminal polar codes for memoryless channels. This is stronger than prior art in which the square root is replaced by a cube root. Our coding scheme is similar yet distinct from prior art. The main differences are: 1) Guard-bands are placed in almost all polarization levels; 2) Trellis decoding is applied to the whole received word, and not to segments of it. As before, the scheme is capacity-achieving. The price we pay for this improvement is a higher decoding complexity, which is nonetheless still polynomial, .
Paper Structure (30 sections, 11 theorems, 112 equations, 12 figures, 1 table, 1 algorithm)

This paper contains 30 sections, 11 theorems, 112 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Preliminaries
  4. Three related channels
  5. Weak polarization for the TDC
  6. Blocks and guard-bands
  7. An overview of this work compared to TPFV:22p
  8. Coding scheme for the Deletion Channel
  9. Encoder
  10. High level decoder description
  11. Trellis operations
  12. Vertices
  13. Edges
  14. Block and GB trellises
  15. Trellis polar transforms
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\delta\in(0,1)$ be a fixed deletion probability of the deletion channel. Fix $\epsilon \in (0,1)$, $\xi \in (0,\frac{1}{6})$, and $0 < \beta' < \beta < \frac{1}{2}$. There exist $n_0^\mathrm{th}(\epsilon,\delta,\xi)$ and $n^\mathrm{th}(\epsilon,\beta,\beta',n_0)$ such that the following holds. is at least $\mathcal{I}-\epsilon$, where $\Lambda$ is the length of $g(\mathbf{X}, n_0,\xi)$. Furt

Figures (12)

  • Figure 1: The random variables $\mathbf{X}$, $\mathbf{G}$, $\mathbf{Y}$, and $\mathbf{Z}$.
  • Figure 2: Recursive trellis transforms.
  • Figure 3: Trellis evolution in the decoder. Panel (a) is the initial trellis $\mathcal{T}$. White rectangles are the block trellises which correspond to the blocks of $\mathbf{X}$. In block trellises, both edges with label '$1$' and edges with label '$0$' exist. The blue rectangles mark the GB trellises, which correspond to the GBs (have only '$0$' labels). The number of sections in each block trellis or GB trellis in $\mathcal{T}$ is the length of the corresponding block or GB. Panel (b) shows the trellis after collapsing each of the GB trellises into a single section and after the first $n_0$ polar transforms. Panel (c) shows how the next polar transform is applied, and is divided into two steps: merging a block trellis with a GB trellis and then merging the resulting trellis with a block trellis (either a '$-$' or a '$+$' merge).
  • Figure 4: An illustration of the prior-art decoder TPFV:22p. The initial step partitions the channel output $\mathbf{y}$ into trimmed outputs of each block. In this example $n=n_0+2$, hence there are four blocks. The next step builds a block trellis for each block. These trellises are then processed according to the successive cancellation decoder, by recursively performing '$-$' and '$+$' transforms to them.
  • Figure 5: Left: The first block trellis $\mathcal{T}^{\mathrm{B-1}}$. Right: The same block trellis after $n_0=2$ minus polar transforms. Red edges are labeled '$1$' and blue edges are labeled '$0$'. To improve legibility, a gray edge represents two parallel edges, red and blue (the two edges do not have the same weight necessarily). In this example $n_0=2$, thus there are four sections in each block trellis.
  • ...and 7 more figures

Theorems & Definitions (17)

  • Theorem 1: Stronger polarization
  • Corollary 2
  • Theorem 3: Weak polarization for the trimmed deletion channel
  • Lemma 4: GBs have a negligible effect on the rate
  • Lemma 5: Recursive bounds for the Bhattacharyya parameter of the TDC
  • Lemma 6: Upper bounding $\sqrt{\mathbb{P}(\neg \mathrm{GBM})}$
  • Lemma 7: Strong polarization despite an additive penalty
  • Claim 6.1
  • Claim 6.2
  • Claim 6.3
  • ...and 7 more