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Achieving uniform side information gain with multilevel lattice codes over the ring of integers

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

TL;DR

The paper develops a CRT lattice index coding framework based on Construction $\\pi_A$ lattices over $\\mathbb{Z}$ to exploit receiver side information in index coding. It derives an explicit upper bound on side information gain for same-rank component codes: $\\Gamma(\\mathcal{C}) \\leq \\frac{n}{k} \\cdot 20 \\log_{10} 2$ dB/bit/dim, and proposes uniform-gain constructions achieving $\\Gamma(\\mathcal{C}) = \\frac{n}{k} \\cdot 20 \\log_{10} 2$ (with asymptotic gains near 6.02 dB/bit/dim as $n$ grows). The work leverages multilevel encoding and the CRT mapping to enable multistage decoding and to relate side-information gains to volume and minimum-distance relations across side-information subsets. It suggests extensions to other rings of integers (e.g., Gaussian, Eisenstein, quaternion orders) to broaden applicability and tighten bounds for index-coding systems.

Abstract

The index coding problem aims to optimise broadcast communication by taking advantage of receiver-side information to improve transmission efficiency. In this letter, we explore the application of Construction $π_A$ lattices to index coding. We introduce a coding scheme, named \textit{CRT lattice index coding}, using Construction $π_A$ over $\mathbb{Z}$ to address the index coding problem. It is derived an upper bound for side information gain of a CRT lattice index code and conditions for the uniformity of this gain. The efficiency of this approach is shown through theoretical analysis and code design examples.

Achieving uniform side information gain with multilevel lattice codes over the ring of integers

TL;DR

The paper develops a CRT lattice index coding framework based on Construction lattices over to exploit receiver side information in index coding. It derives an explicit upper bound on side information gain for same-rank component codes: dB/bit/dim, and proposes uniform-gain constructions achieving (with asymptotic gains near 6.02 dB/bit/dim as grows). The work leverages multilevel encoding and the CRT mapping to enable multistage decoding and to relate side-information gains to volume and minimum-distance relations across side-information subsets. It suggests extensions to other rings of integers (e.g., Gaussian, Eisenstein, quaternion orders) to broaden applicability and tighten bounds for index-coding systems.

Abstract

The index coding problem aims to optimise broadcast communication by taking advantage of receiver-side information to improve transmission efficiency. In this letter, we explore the application of Construction lattices to index coding. We introduce a coding scheme, named \textit{CRT lattice index coding}, using Construction over to address the index coding problem. It is derived an upper bound for side information gain of a CRT lattice index code and conditions for the uniformity of this gain. The efficiency of this approach is shown through theoretical analysis and code design examples.
Paper Structure (9 sections, 3 theorems, 22 equations, 3 figures, 1 table)

This paper contains 9 sections, 3 theorems, 22 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices and lattice codes
  4. Problem Statement
  5. CRT lattice index coding
  6. Main properties
  7. An upper bound for side information gain
  8. CRT lattice index coding with uniform side information gain
  9. Conclusion and Perspectives

Key Result

Proposition 1

For the lattices $\Lambda_1,\ldots, \Lambda_r$ and $\Lambda$ as in (pia_restriction), the following is true, (i) $\text{vol}(\Lambda_{S^c})=\prod_{j\in S} p_j^{k_j}\text{vol}(\Lambda)$; (ii) $d_0\leq d_{S}\leq \prod_{j\in S} p_j d_0$, where $d_S = d_{\min}(\Lambda_{S^c})$ and $d_0 = d_{\min}(\Lambda

Figures (3)

  • Figure 1: Index coding over an AWGN channel with $r$ messages $\{w_1, \ldots, w_r\}$ and $L$ receivers. Each receiver $l$ estimates $(\hat{w}_1^{(l)}, \ldots, \hat{w}_r^{(l)})$ based on their received signal and prior knowledge $w_{S_l} \subset \{w_1, \ldots, w_r\}$, huang2017lattice.
  • Figure 2: SNR versus symbol error rate over the AWGN channel for the code of Example \ref{['exemplo']}.
  • Figure 3: On the left, an example of the CRT lattice index code $\Psi(\mathcal{C}_1 \times \mathcal{C}_2) = \Lambda' / q\mathbb{Z}^2$ where $\mathcal{C}_1 = \langle(1,1)\rangle\subset\mathbb{Z}_2^2$ and $\mathcal{C}_2 = \langle (1,2)\rangle\subset\mathbb{Z}_5^2$. On the right, the associated lattice index code from Gaussian integers, $\varphi(\Lambda_1 / \Lambda' \times \Lambda_2 / \Lambda') = \mathbb{Z}^2 / \Lambda'$, as constructed in natarajan2015lattice.

Theorems & Definitions (12)

  • Definition 1: Construction $\pi_A$ lattice, huang2017construction
  • Proposition 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Example 1
  • Remark 2
  • Theorem 1
  • proof
  • ...and 2 more