Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and Deletions

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. Supported by NSF CAREER Award CNS-2047272 and NSF Awards CCF-1910659 and CNS-1931443, Supported by NSF CCF-1910659 and NSF CCF-1910411. Supported by NSF CAREER Award CCF-1845349 and NSF Award CCF-2127575. Supported by NSF CAREER Award CCF-1845349.

[1]  Bernhard Haeupler,et al.  Synchronization strings: codes for insertions and deletions approaching the Singleton bound , 2017, STOC.

[2]  Venkatesan Guruswami,et al.  Deletion Codes in the High-Noise and High-Rate Regimes , 2014, IEEE Transactions on Information Theory.

[3]  Ronald de Wolf,et al.  A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs , 2007, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[4]  Rafail Ostrovsky,et al.  Private Locally Decodable Codes , 2007, ICALP.

[5]  Sergey Yekhanin,et al.  Towards 3-query locally decodable codes of subexponential length , 2008, JACM.

[6]  Zhengzhong Jin,et al.  Block Edit Errors with Transpositions: Deterministic Document Exchange Protocols and Almost Optimal Binary Codes , 2018, ICALP.

[7]  Venkatesan Guruswami,et al.  Optimally resilient codes for list-decoding from insertions and deletions , 2019, Electron. Colloquium Comput. Complex..

[8]  Bernhard Haeupler,et al.  Near-linear time insertion-deletion codes and (1+ε)-approximating edit distance via indexing , 2018, STOC.

[9]  Michael Mitzenmacher,et al.  A Survey of Results for Deletion Channels and Related Synchronization Channels , 2008, SWAT.

[10]  Venkatesan Guruswami,et al.  Efficient Linear and Affine Codes for Correcting Insertions/Deletions , 2021, SODA.

[11]  Luca Trevisan,et al.  Pseudorandom Generators without the XOR Lemma (Abstract). , 1999 .

[12]  Zhengzhong Jin,et al.  Deterministic Document Exchange Protocols, and Almost Optimal Binary Codes for Edit Errors , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[13]  Arnab Bhattacharyya,et al.  Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs , 2015, ACM Trans. Comput. Theory.

[14]  William I. Gasarch,et al.  A Survey on Private Information Retrieval (Column: Computational Complexity) , 2004, Bull. EATCS.

[15]  Bernhard Haeupler,et al.  Synchronization strings: explicit constructions, local decoding, and applications , 2017, STOC.

[16]  Michael Viderman,et al.  Locally Testable vs. Locally Decodable Codes , 2010, APPROX-RANDOM.

[17]  Xin Li,et al.  Efficient Document Exchange and Error Correcting Codes with Asymmetric Information , 2020, SODA.

[18]  Bernhard Haeupler.  Optimal Document Exchange and New Codes for Insertions and Deletions , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[19]  Richard J. Lipton,et al.  A New Approach To Information Theory , 1994, STACS.

[20]  Rafail Ostrovsky,et al.  Public Key Locally Decodable Codes with Short Keys , 2011, APPROX-RANDOM.

[21]  Amirbehshad Shahrasbi,et al.  Synchronization Strings and Codes for Insertions and Deletions—A Survey , 2021, IEEE Transactions on Information Theory.

[22]  Jonathan Katz,et al.  On the efficiency of local decoding procedures for error-correcting codes , 2000, STOC '00.

[23]  Xin Li,et al.  Synchronization Strings: Highly Efficient Deterministic Constructions over Small Alphabets , 2019, SODA.

[24]  Ronen Shaltiel,et al.  Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels , 2016, APPROX-RANDOM.

[25]  Venkatesan Guruswami,et al.  Coding against deletions in oblivious and online models , 2018, SODA.

[26]  Silvio Micali,et al.  Optimal Error Correction Against Computationally Bounded Noise , 2005, TCC.

[27]  Shubhangi Saraf,et al.  Guest Column: Local Testing and Decoding of High-Rate Error-Correcting Codes , 2016, SIGA.

[28]  Arnab Bhattacharyya,et al.  Lower Bounds for 2-Query LCCs over Large Alphabet , 2017, APPROX-RANDOM.

[29]  Ronald de Wolf.  Error-Correcting Data Structures , 2009, STACS.

[31]  Vahid Tarokh,et al.  A survey of error-correcting codes for channels with symbol synchronization errors , 2010, IEEE Communications Surveys & Tutorials.

[32]  Ron Rothblum,et al.  Relaxed Locally Correctable Codes , 2018, ITCS.

[33]  Venkatesan Guruswami,et al.  Polynomial Time Decodable Codes for the Binary Deletion Channel , 2019, IEEE Transactions on Information Theory.

[34]  Anna Gál,et al.  Three-Query Locally Decodable Codes with Higher Correctness Require Exponential Length , 2012, TOCT.

[35]  Luca Trevisan,et al.  Lower bounds for linear locally decodable codes and private information retrieval , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[36]  Avi Wigderson,et al.  Superquadratic Lower Bound for 3-Query Locally Correctable Codes over the Reals , 2017, Theory Comput..

[37]  Madhu Sudan,et al.  Synchronization Strings: List Decoding for Insertions and Deletions , 2018, ICALP.

[38]  Vladimir I. Levenshtein,et al.  Binary codes capable of correcting deletions, insertions, and reversals , 1965 .

[39]  Shubhangi Saraf,et al.  Locally Decodable Codes , 2016, Encyclopedia of Algorithms.

[40]  David P. Woodruff.  A Quadratic Lower Bound for Three-Query Linear Locally Decodable Codes over Any Field , 2010, Journal of Computer Science and Technology.

[41]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[42]  Yu Zheng,et al.  Locally Decodable Codes with Randomized Encoding , 2020, IACR Cryptol. ePrint Arch..

[43]  Manuel Blum,et al.  Designing programs that check their work , 1989, STOC '89.

[44]  Rafail Ostrovsky,et al.  Local correctability of expander codes , 2013, Inf. Comput..

[45]  Eli Ben-Sasson,et al.  Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding , 2004, SIAM J. Comput..

[46]  Zeev Dvir,et al.  Tight lower bounds for linear 2-query LCCs over finite fields , 2016, Comb..

[47]  M. Talagrand,et al.  Probability in Banach Spaces: Isoperimetry and Processes , 1991 .

[48]  Or Meir,et al.  High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity , 2015, STOC.

[49]  Eyal Kushilevitz,et al.  Private information retrieval , 1998, JACM.

[50]  Luca Trevisan,et al.  Some Applications of Coding Theory in Computational Complexity , 2004, Electron. Colloquium Comput. Complex..

[51]  Samson Zhou,et al.  Relaxed Locally Correctable Codes in Computationally Bounded Channels , 2018, IEEE Transactions on Information Theory.

[52]  Zeev Dvir,et al.  Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits , 2007, SIAM J. Comput..

[53]  Rafail Ostrovsky,et al.  Public-Key Locally-Decodable Codes , 2008, CRYPTO.

[54]  David Zuckerman,et al.  Asymptotically good codes correcting insertions, deletions, and transpositions , 1997, SODA '97.

[55]  Jirí Matousek,et al.  Expected Length of the Longest Common Subsequence for Large Alphabets , 2003, LATIN.

[56]  Ronald de Wolf,et al.  Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval , 2004, ICALP.

[57]  Anat Paskin-Cherniavsky,et al.  Locally Decodable Codes for Edit Distance , 2015, ICITS.

[58]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[59]  David P. Woodruff.  New Lower Bounds for General Locally Decodable Codes , 2007, Electron. Colloquium Comput. Complex..

[60]  R. Dudley.  Central Limit Theorems for Empirical Measures , 1978 .

[61]  Shubhangi Saraf,et al.  High-rate codes with sublinear-time decoding , 2014, Electron. Colloquium Comput. Complex..

[62]  LundCarsten,et al.  Algebraic methods for interactive proof systems , 1992 .

[63]  Oded Lachish,et al.  On the Power of Relaxed Local Decoding Algorithms , 2020, SODA.

[64]  Alexandr Andoni,et al.  Optimal Hashing-based Time-Space Trade-offs for Approximate Near Neighbors , 2016, SODA.

[65]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[66]  Venkatesan Guruswami,et al.  Efficient Low-Redundancy Codes for Correcting Multiple Deletions , 2015, IEEE Transactions on Information Theory.

[67]  Zeev Dvir,et al.  On Matrix Rigidity and Locally Self-correctable Codes , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[68]  Arnab Bhattacharyya,et al.  Combinatorial lower bounds for 3-query LDCs , 2020, ITCS.

[69]  Zeev Dvir,et al.  Matching Vector Codes , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[70]  Jeremiah Blocki,et al.  Private and Resource-Bounded Locally Decodable Codes for Insertions and Deletions , 2021, 2021 IEEE International Symposium on Information Theory (ISIT).

[71]  Klim Efremenko,et al.  3-Query Locally Decodable Codes of Subexponential Length , 2008 .

[72]  Shu Liu,et al.  On List Decoding of Insertion and Deletion Errors , 2019, ArXiv.