Decoding Balanced Linear Codes With Preprocessing
Why this work is in the frame
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Bibliographic record
Abstract
Prange's information set algorithm is a decoding algorithm for arbitrary linear codes. It decodes corrupted codewords of any $\mathbb{F}_2$-linear code $C$ of message length $n$ up to relative error rate $O(\log n / n)$ in $\mathsf{poly}(n)$ time. We show that the error rate can be improved to $O((\log n)^2 / n)$, provided: (1) the decoder has access to a polynomial-length advice string that depends on $C$ only, and (2) $C$ is $n^{-Ω(1)}$-balanced. As a consequence we improve the error tolerance in decoding random linear codes if inefficient preprocessing of the code is allowed. This reveals potential vulnerabilities in cryptographic applications of Learning Noisy Parities with low noise rate. Our main technical result is that the Hamming weight of $Hw$, where $H$ is a random sample of *short dual* codewords, measures the proximity of a word $w$ to the code in the regime of interest. Given such $H$ as advice, our algorithm corrects errors by locally minimizing this measure. We show that for most codes, the error rate tolerated by our decoder is asymptotically optimal among all algorithms whose decision is based on thresholding $Hw$ for an arbitrary polynomial-size advice matrix $H$.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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