Computing the Asymptotic Expected Multiplicity of Elementary Trapping Sets (ETSs) in Random LDPC Code Ensembles
Bibliographic record
Abstract
The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some substructures of the code's Tanner graple (simple) cycle have a nonzero finite average multiplicity. In this paper, we compute the asymptotic expected multiplicity of such ETS structures in random LDPC code ensembles. The computation, in general, involves two subproblems: (i) counting the number of cycles of a certain length with different combinations of variable node degrees, and (ii) counting the number of ways trees of different sizes can be appended to such a cycle to form an ETS. The first subproblem, which counts the number of leafless ETSs (LETSs), is solved by the authors in other papers. The second subproblem is formulated as a recursive counting problem and is solved in this paper, with the solution being a generalization of Catalan numbers. We also demonstrate through numerical results that the asymptotic expected values computed in this paper match the multiplicity of ETSs in randomly selected finite-length LDPC codes, even at relatively short block lengths.
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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 | 0.001 | 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 | 0.000 |
| 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 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".