Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gates.
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Bibliographic record
Abstract
We bound the minimum number w of wires needed to compute any (asymptotically good) error-correcting code C: {0, 1} Ω(n) → {0, 1} n with minimum distance Ω(n), using unbounded fan-in circuits of depth d with arbitrary gates. Our main results are: (1) If d = 2 then w = Θ(n(lg n / lg lg n) 2). (2) If d = 3 then w = Θ(n lg lg n). (3) If d = 2k or d = 2k + 1 for some integer k ≥ 2 then w = Θ(nλk(n)), where λ1(n) = ⌈lg n⌉, λi+1(n) = λ ∗ i (n), and the ∗ operation gives how many times one has to iterate the function λi to reach a value at most 1 from the argument n. (4) If d = lg ∗ n then w = O(n). For depth d = 2, our Ω(n(lg n / lg lg n) 2) lower bound gives the largest known lower bound for computing any linear map. Using a result by Ishai, Kushilevitz, Ostrovsky, and Sahai [17], we also obtain similar bounds for computing pairwise-independent hash functions. Our lower bounds are based on a superconcentrator-like condition that the graphs of circuits computing good codes must satisfy. This condition is provably intermediate between superconcentrators and their weakenings considered before.
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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 | 0.000 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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