Locally Decodable Codes (LDCs) are errorcorrecting 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 3query 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 PaskinCherniavsky (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 2query linear Insdel LDCs do not exist, and give an exponential lower bound for the length of all qquery 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 privatekey 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 privatekey 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.
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This content will become publicly available on July 1, 2024
A NearCubic Lower Bound for 3Query Locally Decodable Codes from Semirandom CSP Refutation
A code C ∶ {0,1}k → {0,1}n is a qlocally decodable code (qLDC) if one can recover any chosen bit bi of the message b ∈ {0,1}k with good confidence by randomly querying the encoding x = C(b) on at most q coordinates. Existing constructions of 2LDCs achieve n = exp(O(k)), and lower bounds show that this is in fact tight. However, when q = 3, far less is known: the best constructions achieve n = exp(ko(1)), while the best known results only show a quadratic lower bound n ≥ Ω(k2/log(k)) on the blocklength.
In this paper, we prove a nearcubic lower bound of n ≥ Ω(k3/log6(k)) on the blocklength of 3query LDCs. This improves on the best known prior works by a polynomial factor in k. Our proof relies on a new connection between LDCs and refuting constraint satisfaction problems with limited randomness. Our quantitative improvement builds on the new techniques for refuting semirandom instances of CSPs and, in particular, relies on bounding the spectral norm of appropriate Kikuchi matrices.
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 Award ID(s):
 2211971
 NSFPAR ID:
 10435079
 Date Published:
 Journal Name:
 ACM Symposium on Theory of Computing, STOC
 Issue:
 2023
 Page Range / eLocation ID:
 1438 to 1448
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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