Relaxed vs. Full Local Decodability with Few Queries: Equivalence and Separations for Linear Codes
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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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 | 0.004 | 0.030 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.004 | 0.014 |
| Open science | 0.003 | 0.007 |
| Research integrity | 0.003 | 0.008 |
| Insufficient payload (model declined to judge) | 0.008 | 0.002 |
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".