On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors

Authors Alexander R. Block , Jeremiah Blocki , Kuan Cheng , Elena Grigorescu , Xin Li , Yu Zheng, Minshen Zhu

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Author Details

Alexander R. Block
  • University of Maryland, College Park, MD, USA
  • Georgetown University, Washington, D.C., USA
Jeremiah Blocki
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Kuan Cheng
  • Center on Frontiers of Computing Studies, Peking University, China
Elena Grigorescu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Xin Li
  • Department of Computer Science, Johns Hopkins University, Baltimore, MD, USA
Yu Zheng
  • Meta Platforms, Inc., Bellevue, WA, USA
Minshen Zhu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA


We are indebted to some anonymous reviewers who helped us improve the presentation of the paper.

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Alexander R. Block, Jeremiah Blocki, Kuan Cheng, Elena Grigorescu, Xin Li, Yu Zheng, and Minshen Zhu. On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 14:1-14:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Locally Decodable Codes (LDCs) are error-correcting codes C:Σⁿ → Σ^m, encoding messages in Σⁿ to codewords in Σ^m, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length m that is super-polynomial in n, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Standard LDCs for Insdel errors were first studied by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), and are further motivated by recent advances in DNA random access bio-technologies. Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries (even adaptively), over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a "phase-transition"-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits. We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with almost linear codeword length and constant query complexity, matching the parameters of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Mathematics of computing → Coding theory
  • Theory of computation → Lower bounds and information complexity
  • Relaxed Locally Decodable Codes
  • Hamming Errors
  • Insdel Errors
  • Lower Bounds


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