Constant Query Local Decoding Against Deletions Is Impossible

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Locally decodable codes (LDC's) are error-correcting codes that allow recovery of individual message indices by accessing only a constant number of codeword indices. For substitution errors, it is evident that LDC's exist -- Hadamard codes are examples of 2-query LDC's. Research on this front has focused on finding the optimal encoding length for LDC's, for which there is a nearly exponential gap between the best lower bounds and constructions.
Ostrovsky and Paskin-Cherniavsky (ICITS 2015) introduced the notion of local decoding to the insertion and deletion setting. In this context, it is not clear whether constant query LDC's exist at all. Indeed, in contrast to the classical setting, Block et al. conjecture that they do not exist. Blocki et al. (FOCS 2021) make progress towards this conjecture, proving that any potential code must have at least exponential encoding length.
Our work definitively resolves the conjecture and shows that constant query LDC's do not exist in the insertion/deletion (or even deletion-only) setting. Using a reduction shown by Blocki et al., this also implies that constant query locally correctable codes do not exist in this setting.