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Multilevel lattice codes from Hurwitz quaternion integers

Juliana G. F. Souza, Sueli I. R. Costa, Cong Ling

TL;DR

The paper develops a multilevel lattice coding framework by extending Construction pi_A to Hurwitz quaternion integers via CRT-based isomorphisms on maximal orders, enabling efficient multistage decoding and achieving the Poltyrev-limit. It provides a concrete Hurwitz-based construction, analyzes decoding complexity, and proves Poltyrev-goodness through averaging over balanced code ensembles. An index coding application demonstrates the practical benefits and flexibility of the Hurwitz-based construction. The work also outlines promising future directions, including extensions to other maximal orders and to octonions, with potential impact on advanced communication paradigms like compute-and-forward.

Abstract

This work presents an extension of the Construction $π_A$ lattices proposed in \cite{huang2017construction}, to Hurwitz quaternion integers. This construction is provided by using an isomorphism from a version of the Chinese remainder theorem applied to maximal orders in contrast to natural orders in prior works. Exploiting this map, we analyze the performance of the resulting multilevel lattice codes, highlight via computer simulations their notably reduced computational complexity provided by the multistage decoding. Moreover it is shown that this construction effectively attain the Poltyrev-limit.

Multilevel lattice codes from Hurwitz quaternion integers

TL;DR

The paper develops a multilevel lattice coding framework by extending Construction pi_A to Hurwitz quaternion integers via CRT-based isomorphisms on maximal orders, enabling efficient multistage decoding and achieving the Poltyrev-limit. It provides a concrete Hurwitz-based construction, analyzes decoding complexity, and proves Poltyrev-goodness through averaging over balanced code ensembles. An index coding application demonstrates the practical benefits and flexibility of the Hurwitz-based construction. The work also outlines promising future directions, including extensions to other maximal orders and to octonions, with potential impact on advanced communication paradigms like compute-and-forward.

Abstract

This work presents an extension of the Construction lattices proposed in \cite{huang2017construction}, to Hurwitz quaternion integers. This construction is provided by using an isomorphism from a version of the Chinese remainder theorem applied to maximal orders in contrast to natural orders in prior works. Exploiting this map, we analyze the performance of the resulting multilevel lattice codes, highlight via computer simulations their notably reduced computational complexity provided by the multistage decoding. Moreover it is shown that this construction effectively attain the Poltyrev-limit.
Paper Structure (14 sections, 9 theorems, 84 equations, 2 figures, 4 tables, 1 algorithm)

This paper contains 14 sections, 9 theorems, 84 equations, 2 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Lattices and lattice codes
  5. Quaternion Algebras
  6. Multilevel lattice codes over Hurwitz integers
  7. Multilevel Decoding Process
  8. Decoding complexity
  9. Goodness for channel coding
  10. Balanced set of codes over rings
  11. Balanced set of codes from the Chinese Remainder Theorem
  12. Poltyrev-goodness
  13. Application to Lattice Index Coding
  14. Conclusion and Perspectives

Key Result

Proposition 1

For all $\alpha, \beta\in \mathscr{H}$ not both zero, there exist $\mu, \nu\in\mathscr{H}$ such that $\mu\alpha+\nu\beta = \delta$, where $\delta$ is a left greatest common divisor of $\alpha$ and $\beta$.

Figures (2)

  • Figure 1: Projections of the codes of Example \ref{['exemplo_pia_quaternio']} onto $\mathbb{R}^3$ by omitting the real part. Gray points represent the projection of all 81 elements of $\mathscr{H}/3\mathscr{H}$. In (a), the code $\mathcal{C}_1\subset\mathscr{H}/\mathscr{H}\pi$ is shown in blue; in (b), the code $\mathcal{C}_2\subset\mathscr{H}/\mathscr{H}\overline{\pi}$ is shown in red, and, in (c) the code $\mathcal{C}=\varphi^{-1}(\mathcal{C}_1\times\mathcal{C}_2)\subset\mathscr{H}/3\mathscr{H}$, obtained via Construction $\pi_A$, is represented by black points. The cube serves as a spatial reference in $\mathbb{R}^3$.
  • Figure 2: Comparison of decoding complexity for constellations with cardinality $q^4$ in $\mathbb{R}^4$ obtained through Construction $\pi_A$ over $\mathbb{Z}[i]$, $\mathbb{Z}[\omega]$, and $\mathscr{H}$. Here, $\mathcal{C}_{\text{max}}$ denotes the maximum cardinality among the layers in the encoding.

Theorems & Definitions (29)

  • Definition 1: Construction A
  • Definition 2: Quaternion algebra
  • Definition 3: Order
  • Example 1
  • Definition 4: Left ideal
  • Definition 5: Left prime ideal, reyes2010one
  • Definition 6: Prime ideal, reiner1975maximal
  • Definition 7: Maximal Order
  • Example 2
  • Definition 8: Hurwitz quaternion integers
  • ...and 19 more