Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and Deletions
This work investigates the existence of locally decodable codes (LDCs) under insertion-deletion (insdel) errors. Addressing a long-standing open conjecture, we prove—*for the first time*—that no 2-query linear insdel LDC exists. Moreover, for any constant query complexity $q geq 3$, we establish an exponential lower bound $exp(Omega(n))$ on the code length, significantly stronger than the polynomial bounds known for Hamming-error LDCs. Our approach constructs a hard insdel error distribution and combines information-theoretic analysis with novel coding reduction techniques. This reveals a fundamental separation between insdel LDCs and Hamming LDCs—a separation that persists even in the adaptive decoding and private-key settings. The results characterize the theoretical limits of local error correction against synchronization errors and provide the first tight lower bounds for insdel coding.