Relaxed vs. Full Local Decodability with Few Queries: Equivalence and Separations for Linear Codes
Why this work is in the frame
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Bibliographic record
Abstract
A locally decodable code (LDC) $C \colon \{0,1\}^k \to \{0,1\}^n$ is an error-correcting code that allows one to recover any bit of the original message with good probability while only reading a small number of bits from a corrupted codeword. A relaxed locally decodable code (RLDC) is a weaker notion where the decoder is additionally allowed to abort and output a special symbol $\bot$ if it detects an error. For a large constant number of queries $q$, there is a large gap between the blocklength $n$ of the best $q$-query LDC and the best $q$-query RLDC. Existing constructions of RLDCs achieve polynomial length $n = k^{1 + O(1/q)}$, while the best-known $q$-LDCs only achieve subexponential length $n = 2^{k^{o(1)}}$. On the other hand, for $q = 2$, it is known that RLDCs and LDCs are equivalent. We thus ask the question: what is the smallest $q$ such that there exists a $q$-RLDC that is not a $q$-LDC? In this work, we show that any linear $3$-query RLDC is in fact a $3$-LDC, i.e., linear RLDCs and LDCs are equivalent at $3$ queries. More generally, we show for any constant $q$, there is a soundness error threshold $s(q)$ such that any linear $q$-RLDC with soundness error below this threshold must be a $q$-LDC. This implies that linear RLDCs cannot have "strong soundness" -- a stricter condition satisfied by linear LDCs that says the soundness error is proportional to the fraction of errors in the corrupted codeword -- unless they are simply LDCs. In addition, we give simple constructions of linear $15$-query RLDCs that are not $q$-LDCs for any constant $q$, showing that for $q = 15$, linear RLDCs and LDCs are not equivalent. We also prove nearly identical results for locally correctable codes and their corresponding relaxed counterpart.
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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 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| 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