Problem
Research questions and friction points this paper is trying to address.
Proves exponential lower bounds for Insdel LDC length
Shows strict separation between Hamming and Insdel LDCs
Establishes non-existence of 2-query linear Insdel LDCs
Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and Deletions
🤖 AI Summary
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.
📝 Abstract
Locally Decodable Codes (LDCs) are error-correcting codes for which individual message symbols can be quickly recovered despite errors in the codeword. LDCs for Hamming errors have been studied extensively in the past few decades, where a major goal is to understand the amount of redundancy that is necessary and sufficient to decode from large amounts of error, with small query complexity. Despite exciting progress, we still don't have satisfactory answers in several important parameter regimes. For example, in the case of 3-query LDCs, the gap between existing constructions and lower bounds is superpolynomial in the message length. In this work we study LDCs for insertion and deletion errors, called Insdel LDCs. Their study was initiated by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), who gave a reduction from Hamming LDCs to Insdel LDCs with a small blowup in the code parameters. On the other hand, the only known lower bounds for Insdel LDCs come from those for Hamming LDCs, thus there is no separation between them. Here we prove new, strong lower bounds for the existence of Insdel LDCs. In particular, we show that 2-query linear Insdel LDCs do not exist, and give an exponential lower bound for the length of all q-query Insdel LDCs with constant q. For $q$ ≥ 3 our bounds are exponential in the existing lower bounds for Hamming LDCs. Furthermore, our exponential lower bounds continue to hold for adaptive decoders, and even in private-key settings where the encoder and decoder share secret randomness. This exhibits a strict separation between Hamming LDCs and Insdel LDCs. Our strong lower bounds also hold for the related notion of Insdel LCCs (except in the private-key setting), due to an analogue to the Insdel notions of a reduction from Hamming LCCs to LDCs. Our techniques are based on a delicate design and analysis of hard distributions of insertion and deletion errors, which depart significantly from typical techniques used in analyzing Hamming LDCs.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Exponential lower bounds for Insdel LDCs
Hard distributions for insertion and deletion errors
Strict separation from Hamming LDCs
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