For every odd q ≥ 3, any q-query binary locally decodable code with constant distance satisfies k ≤ O~(n^(1-2/q)), the first bound of this form for q ≥ 5.
Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes
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Improved Lower Bounds for all Odd-Query Locally Decodable Codes
For every odd q ≥ 3, any q-query binary locally decodable code with constant distance satisfies k ≤ O~(n^(1-2/q)), the first bound of this form for q ≥ 5.