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Revisit the AWGN-goodness of Polar-like Lattices

Ling Liu, Junjiang Yu, Shanxiang Lyu, Baoming Bai

TL;DR

This work studies polar lattices and PAC lattices from a generator-matrix viewpoint and proves AWGN-goodness for the polar and PAC lattice families under multilevel decoding. It shows how to construct polar lattices via lifting and row scaling of the polar code generator and analyzes their volume and decoding complexity, establishing conditions (r = O(log N), C(Z, σ^2) → 0) under which the normalized volume-to-noise ratio converges to the AWGN limit. The PAC lattice framework is then tied to polar lattices through bar{G}_{pac} = bar{T} × bar{G}, explaining distance and decoding advantages, and highlighting a gluing-theory interpretation of the design. Overall, the paper provides a constructive, scalable approach to achieving AWGN-optimal performance with polar-like lattices and offers design principles to further enhance practical performance via PAC structures and multiplier matrices T.

Abstract

This paper aims to provide a comprehensive introduction to lattices constructed based on polar-like codes and demonstrate some of their key properties, such as AWGN goodness. We first present polar lattices directly from the perspective of their generator matrix. Next, we discuss their connection with the recently proposed PAC (polarization adjusted convolutional) lattices and analyze the structural advantages of PAC lattices, through which the AWGN-goodness of PAC lattices can be conveniently demonstrated.

Revisit the AWGN-goodness of Polar-like Lattices

TL;DR

This work studies polar lattices and PAC lattices from a generator-matrix viewpoint and proves AWGN-goodness for the polar and PAC lattice families under multilevel decoding. It shows how to construct polar lattices via lifting and row scaling of the polar code generator and analyzes their volume and decoding complexity, establishing conditions (r = O(log N), C(Z, σ^2) → 0) under which the normalized volume-to-noise ratio converges to the AWGN limit. The PAC lattice framework is then tied to polar lattices through bar{G}_{pac} = bar{T} × bar{G}, explaining distance and decoding advantages, and highlighting a gluing-theory interpretation of the design. Overall, the paper provides a constructive, scalable approach to achieving AWGN-optimal performance with polar-like lattices and offers design principles to further enhance practical performance via PAC structures and multiplier matrices T.

Abstract

This paper aims to provide a comprehensive introduction to lattices constructed based on polar-like codes and demonstrate some of their key properties, such as AWGN goodness. We first present polar lattices directly from the perspective of their generator matrix. Next, we discuss their connection with the recently proposed PAC (polarization adjusted convolutional) lattices and analyze the structural advantages of PAC lattices, through which the AWGN-goodness of PAC lattices can be conveniently demonstrated.
Paper Structure (10 sections, 5 theorems, 26 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 5 theorems, 26 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries of Polar Codes and Polar Lattices
  3. Polar Codes
  4. Lattice Codes and Polar Lattices
  5. The AWGN-goodness of Polar Lattices
  6. The encoding process
  7. The multilevel decoding process
  8. The AWGN-goodness of polar lattices: revisited
  9. The Polarization Adjusted Convolutional (PAC) Lattices
  10. Conclusion

Key Result

Lemma 1

For an $N$-dimensional polar lattice $\Lambda$ with rate profile $K_\ell$ from $\ell=1$ to $r$, its volume is given by

Figures (5)

  • Figure 1: The $\mathbb{Z}/2\mathbb{Z}$ channel with noise variance $\sigma^2$.
  • Figure 2: The decoding ordering of the coordinate array of $\bm{\lambda}$. The bits decoded at each level is denoted by different symbol. The ordering in this example is $* \to \circ \to \square\to\vartriangle\to\triangledown\to\lozenge\to\cdots\to\boxtimes\to\blacksquare$.
  • Figure 3: The multilevel SC decoding of polar lattices.
  • Figure 4: A random generated upper-triangle matrix.
  • Figure 5: The three lattices $\mathbb{Z}^2$, $\bar{\mathbf{T}}$ and $A_2$.

Theorems & Definitions (20)

  • Definition 1
  • Definition 2: Polar Lattices
  • Example 1
  • Example 2
  • Lemma 1
  • Remark 1
  • Definition 3
  • Example 3
  • Remark 2
  • Definition 4: AWGN-good lattices
  • ...and 10 more