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Linear Insertion Deletion Codes in the High-Noise and High-Rate Regimes

Authors Kuan Cheng , Zhengzhong Jin, Xin Li , Zhide Wei, Yu Zheng

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Author Details

Kuan Cheng
  • Department of Computer Science, Peking University, Beijing, China
Zhengzhong Jin
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Xin Li
  • Department of Computer Science, Johns Hopkins University, Baltimore, MD, USA
Zhide Wei
  • Department of Computer Science, Peking University, Beijing, China
Yu Zheng
  • Meta Platforms Inc

Funding

  • Cheng, Kuan: Supported by a start-up fund of Peking University.
  • Jin, Zhengzhong: Supported in part by DARPA under Agreement No. HR00112020023 and by an NSF grant CNS-2154149. Most of the work was done while the author was a PhD student at Johns Hopkins University, and supported by NSF CAREER Award CCF-1845349.
  • Li, Xin: Supported by NSF CAREER Award CCF-1845349 and NSF Award CCF-2127575.
  • Wei, Zhide: Supported by a start-up fund of Peking University.
  • Zheng, Yu: Most of the work was done while the author was a PhD student at Johns Hopkins University, and partially supported by NSF Award CCF-2127575.

Cite As Get BibTex

Kuan Cheng, Zhengzhong Jin, Xin Li, Zhide Wei, and Yu Zheng. Linear Insertion Deletion Codes in the High-Noise and High-Rate Regimes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.41

Abstract

This work continues the study of linear error correcting codes against adversarial insertion deletion errors (insdel errors). Previously, the work of Cheng, Guruswami, Haeupler, and Li [Kuan Cheng et al., 2021] showed the existence of asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, or achieve rate arbitrarily close to 1/2 even over the binary alphabet. As shown in [Kuan Cheng et al., 2021], these bounds are also the best possible. However, known explicit constructions in [Kuan Cheng et al., 2021], and subsequent improved constructions by Con, Shpilka, and Tamo [Con et al., 2022] all fall short of meeting these bounds. Over any constant size alphabet, they can only achieve rate < 1/8 or correct < 1/4 fraction of errors; over the binary alphabet, they can only achieve rate < 1/1216 or correct < 1/54 fraction of errors. Apparently, previous techniques face inherent barriers to achieve rate better than 1/4 or correct more than 1/2 fraction of errors.
In this work we give new constructions of such codes that meet these bounds, namely, asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, and binary asymptotically good linear insdel codes that can achieve rate arbitrarily close to 1/2. All our constructions are efficiently encodable and decodable. Our constructions are based on a novel approach of code concatenation, which embeds the index information implicitly into codewords. This significantly differs from previous techniques and may be of independent interest. Finally, we also prove the existence of linear concatenated insdel codes with parameters that match random linear codes, and propose a conjecture about linear insdel codes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Error correcting code
  • Edit distance
  • Pseudorandomness
  • Derandomization

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References

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