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.

翻译：局部可解码码（LDC）是一类纠错码，允许通过仅访问恒定数量的码字索引来恢复单个消息索引。针对替代错误，LDC显然存在——哈达玛码是2查询LDC的实例。该领域的研究聚焦于寻找LDC的最优编码长度，而在最优下界与构造之间仍存在近指数级的差距。Ostrovsky与Paskin-Cherniavsky（ICITS 2015）将局部解码的概念引入插入与删除场景。在此背景下，常数查询LDC是否存在尚未明确。事实上，与经典情形相反，Block等人推测其不存在。Blocki等人（FOCS 2021）通过证明任何潜在编码必须具有至少指数级编码长度，推进了该猜想的验证。本工作最终解决了该猜想，证明在插入/删除（甚至仅删除）场景中常数查询LDC不存在。通过Blocki等人所示的归约方法，这一结论亦意味着该场景中不存在常数查询局部可校正码。