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Are there any known constructions of binary locally testable codes with very low (e.g., independent of the length of the codeword) query complexity and "good" rate (e.g., mapping strings of length $k$ to strings of length $k^{1+c}$, for a small constant $c$) that are also locally decodable (even if the query complexity for decoding is very large (but still sublinear))?

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