On the Girth of Quasi-Cyclic Protograph LDPC Codes
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Bibliographic record
Abstract
In this paper, we study the relationships between the girth of the Tanner graph of a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code, the lifting degree, and the size and the structure of the base graph. As a result, for a given base graph, we derive a lower bound on the lifting degree as a necessary condition for the lifted graph to have a certain girth. This also provides an upper bound on the girth of the family of graphs lifted from a given base graph with a given lifting degree. The upper bounds derived here, which are applicable to both regular and irregular base graphs with no parallel edges, are in some cases more general and in some other cases tighter than the existing bounds. The results presented in this work can be used to design cyclic liftings with relatively small degree and relatively large girth. As an example, we present new QC protograph LDPC code constructions with girth 8 using fully connected base graphs. These constructions provide upper bounds on the lifting degree required for achieving girth 8 using fully connected base graphs.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.001 | 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 it