Full-Diversity Construction-D Lattices: Design and Decoding Perspective on Block-Fading Channels

TL;DR

This work addresses reliable communication over block-fading channels by developing algebraic Construction-D lattices defined over rings of integers of number fields. It introduces a semi-systematic generator matrix and a segmentation-based decoding algorithm that achieves full diversity with linear-in-dimension complexity. Empirical results show significant FER gains over Construction-A lattices in diversity-2 settings, while highlighting that gains may be limited to two-level/low-diversity configurations due to error propagation in higher layers. The framework links algebraic number theory with multilevel lattice coding to enable structured, high-performance coding for BF channels with potential practical impact in wireless reliability.

Abstract

This paper introduces a novel framework for constructing algebraic lattices based on Construction-D, leveraging nested linear codes and prime ideals from algebraic number fields. We focus on the application of these lattices in block-fading (BF) channels, which are characterized by piecewise-constant fading across blocks of transmitted symbols. This approach results in a semi-systematic generator matrix, providing a structured foundation for high-dimensional lattice design for BF channels. The proposed Construction-D lattices exhibit the full diversity property, making them highly effective for error performance improvement. To address this, we develop an efficient decoding algorithm designed specifically for full-diversity Construction-D lattices. Simulations indicate that the proposed lattices notably enhance error performance compared to full-diversity Construction-A lattices in diversity-2 cases. Interestingly, unlike AWGN channels, the expected performance enhancement of Construction-D over Construction-A, resulting from an increased number of nested code levels, was observed only in the two-level and diversity-2 cases. This phenomenon is likely attributed to the intensified effects of error propagation that occur during successive cancellation at higher levels, as well as the higher diversity orders. These findings highlight the promise of Construction-D lattices as an effective coding strategy for enhancing communication reliability in BF channels.

Figures (5)

• Figure 1: The generator matrix of the Construction-D lattice $\Lambda$.
• Figure 2: Poltyrev outage limit for full-diversity Construction-D lattices with $[N,k_2,k_1]=[100,50,40]$ and different diversity orders.
• Figure 3: Decoding of diversity-2 Construction-D lattices with $m=2, 7, 10$.
• Figure 4: Decoding of diversity-2 Construction-D lattices over $\mathbb{Q}(\sqrt{2})$ for $a=2,3,4$.
• Figure 5: Decoding of diversity-3 Construction-D lattices over the number field $\mathbb{Q}(\nu)$ with discriminant $d_K=148$.

Theorems & Definitions (19)

• Definition 1
• Theorem 2.1: 485720
• Definition 2
• Definition 3
• Definition 4: Monogenic Number Fields
• Definition 5: Quadratic Number Fields
• Theorem 2.2: Alaca
• Theorem 2.3
• Definition 6
• Theorem 3.1