Improved Lower Bounds for the Error Rate of Linear Block Codes
Bibliographic record
Abstract
We obtain two lower bounds on the error rate of linear binary block codes (under maximum likelihood decoding) over BPSK-modulated AWGN channels. We cast the problem of finding a lower bound on the probability of a union as an optimization problem which seeks to find the subset which maximizes a recent lower bound – due to Kuai, Alajaji, and Takahara – that we will refer to as the KAT bound. Two variations of the KAT lower bound are then derived. The first bound, the LB-f bound, requires the weight of the product of the codewords with minimum weight in addition to their weight enumeration, while the other bound, the LB-s bound (which is the main contribution of this paper), is algorithmic and only needs the weight enumeration function of the code. The use of a subset of the codebook to evaluate the KAT lower bound not only reduces computational complexity, but also tightens this bound specially at low signal-to-noise (SNR) ratios. Numerical results for binary block codes indicate that at low SNRs the LB-f bound is tighter than the LB-s bound. At high SNRs, the LB-s bound is tighter than other recent lower bounds in the literature, which comprise the lower bound due to Seguin, the original KAT bound (evaluated on the entire codebook), and the dot-product and norm bounds due to Cohen and Merhav.
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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 | 0.008 | 0.044 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.003 | 0.005 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.016 | 0.005 |
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".