Full-Diversity Construction-D Lattices: Design and Decoding Perspective on Block-Fading Channels
Bibliographic record
Abstract
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"><sub>p</sub><sup>N</sup></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<sub>min</sub></i>(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C<sub>a</sub></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.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".