Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of Simultaneously Good Polar Codes and Polar Lattices

Ling Liu, Ruimin Yuan, Shanxiang Lyu, Cong Ling, Baoming Bai

TL;DR

This work addresses constructing objects that are simultaneously good for channel coding and source coding. It introduces a polarization-based chaining framework to build simultaneously good polar codes, then extends the construction to multilevel lattices via Construction D to yield polar lattices that are both AWGN-good and quantization-good. The key contributions include an explicit pairing of two polar codes with identical length and rate, a chained-long-code design with provable error and distortion bounds, and a multilevel polar-lattice construction with theoretical guarantees $\,\lim_{N\to\infty} G(\,\hat{\Lambda}) = 1/(2\pi e)$. The practical impact lies in bridging channel coding, source coding, and shaping through polar techniques, enabling lattice-based schemes with quantization and modulation optimality.

Abstract

In this work, we investigate the simultaneous goodness of polar codes and polar lattices. The simultaneous goodness of a lattice or a code means that it is optimal for both channel coding and source coding simultaneously. The existence of such kind of lattices was proven by using random lattice ensembles. Our work provides an explicit construction based on the polarization technique.

Construction of Simultaneously Good Polar Codes and Polar Lattices

TL;DR

This work addresses constructing objects that are simultaneously good for channel coding and source coding. It introduces a polarization-based chaining framework to build simultaneously good polar codes, then extends the construction to multilevel lattices via Construction D to yield polar lattices that are both AWGN-good and quantization-good. The key contributions include an explicit pairing of two polar codes with identical length and rate, a chained-long-code design with provable error and distortion bounds, and a multilevel polar-lattice construction with theoretical guarantees . The practical impact lies in bridging channel coding, source coding, and shaping through polar techniques, enabling lattice-based schemes with quantization and modulation optimality.

Abstract

In this work, we investigate the simultaneous goodness of polar codes and polar lattices. The simultaneous goodness of a lattice or a code means that it is optimal for both channel coding and source coding simultaneously. The existence of such kind of lattices was proven by using random lattice ensembles. Our work provides an explicit construction based on the polarization technique.
Paper Structure (9 sections, 5 theorems, 27 equations, 3 figures)

This paper contains 9 sections, 5 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries of Polar Lattices
  3. Polar Codes for Channel and Source Coding
  4. Lattice Codes and Polar Lattices
  5. Simultaneously Good Polar Codes
  6. Two Good Codes with Same Rate
  7. Chaining to a Single Long Code
  8. Simultaneously Good Polar Lattices
  9. Conclusion

Key Result

Lemma 1

For any small $\delta>0$ and $0<p<p+\delta<\frac{1}{2}$, there exist two good polar codes $\mathscr{P}_1$ and $\mathscr{P}_2$ with the same block length $N$ and rate $I(V)<\frac{K}{N} < I(W)$ such that $\mathscr{P}_1$ guarantees a decoding error probability less than $N2^{-N^{\beta}}$ as a channel c

Figures (3)

  • Figure 1: The constructions of $\mathscr{P}_1$ and $\mathscr{P}_2$ in Lemma \ref{['lem:2good']}.
  • Figure 2: The structure of the chained polar code $\widehat{\mathscr{P}}$. The set partition marked in red is for channel coding, while the partition in green is for source coding. The encoding for channel coding is performed from the left to the right, with the block index decreases from $B$ to $1$. The set $\mathtt{i}_c^1$ in the 1-st block is fed with frozen bits instead of message bits; The encoding for source coding is performed reversely from Block 1 to $B$. For the last block, the set $\mathtt{i}_s^B$ is not determined by the source vector, but is fed with frozen bits. The arrows between the shadowed blocks highlight these special treatments.
  • Figure 3: The average extra distortion caused by $\mathtt{i}_s^B$ with different sizes.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3: AWGN-good lattices
  • Definition 4: Quantization-good lattices
  • Definition 5
  • Lemma 1
  • Remark 1
  • Lemma 2
  • Definition 6
  • Theorem 1
  • ...and 4 more