Computing the Asymptotic Expected Multiplicity of Leafless Elementary Trapping Sets (LETSs) in Random Irregular LDPC Code Ensembles
Bibliographic record
Abstract
Leafless elementary trapping sets (LETSs) are known to be among the most harmful substructures of Tanner graphs of low-density parity-check (LDPC) codes in the error floor region. It is well-known that in the asymptotic regime of infinite block length, the only LETS structures that have finite non-zero average multiplicity are those that contain a single (simple) cycle. The average multiplicity of all the other LETSs, whose structures contain more than one cycle, tends to zero asymptotically. The asymptotic average multiplicity of LETS structures with a single cycle for variable-regular LDPC codes has already been derived by the authors in a previous work. In this paper, we compute the asymptotic average multiplicity of such structures for irregular LDPC code ensembles. The derivation involves counting the average number of cycles of a specific length with a specific combination of variable node degrees. We demonstrate that the asymptotic results, derived in this paper, match well with the LETS multiplicities of finite-length LDPC codes chosen randomly from the ensemble, even for rather short block lengths.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.014 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 source (direct Gemma or distilled Codex), 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".