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

This paper presents a novel framework for constructing full-diversity algebraic lattices based on Construction-D, utilizing nested linear codes <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ⊂ · · · ⊂ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>a</i></sub> ⊆ F<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_p^N</i> and prime ideals from algebraic number fields of degree n. Focused on block-fading (BF) channels, this approach yields a semi-systematic generator matrix suited for high-dimensional lattice design. The resulting Construction-D lattices achieve full diversity, significantly enhancing error performance. Additionally, we develop a decoding algorithm tailored for these full-diversity lattices, achieving linear complexity relative to the lattice dimension. 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 due to the compounded error propagation during successive cancellation at higher levels, further amplified by higher diversity orders. The omission of parity-check enforcement at the final code level—aimed at simplifying our decoding—restricts the diversity order from the maximum of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>·<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d_min</i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_a</i>) to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>, limiting performance gains. Unraveling the core reasons behind this decoding behavior remains an open challenge. Nevertheless, these findings highlight the promise of full-diversity Construction-D lattices as an effective coding strategy for BF channels.