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April 2020 Lower bounds for trace reconstruction
Nina Holden, Russell Lyons
Ann. Appl. Probab. 30(2): 503-525 (April 2020). DOI: 10.1214/19-AAP1506

Abstract

In the trace reconstruction problem, an unknown bit string ${\mathbf{x}}\in\{0,1\}^{n}$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string ${\widetilde{\mathbf{x}}}$. How many i.i.d. samples of ${\widetilde{\mathbf{x}}}$ are needed to reconstruct ${\mathbf{x}}$ with high probability? We prove that there exist ${\mathbf{x}},{\mathbf{y}}\in\{0,1\}^{n}$ such that at least $cn^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\mathbf{x}}$ and ${\mathbf{y}}$ for some absolute constant $c$, improving the previous lower bound of $cn$. Furthermore, our result improves the previously known lower bound for reconstruction of random strings from $c\log^{2}n$ to $c\log^{9/4}n/\sqrt{\log\log n}$.

Citation

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Nina Holden. Russell Lyons. "Lower bounds for trace reconstruction." Ann. Appl. Probab. 30 (2) 503 - 525, April 2020. https://doi.org/10.1214/19-AAP1506

Information

Received: 1 August 2018; Revised: 1 May 2019; Published: April 2020
First available in Project Euclid: 8 June 2020

zbMATH: 07236126
MathSciNet: MR4108114
Digital Object Identifier: 10.1214/19-AAP1506

Subjects:
Primary: 62C20, 68Q25, 68W32
Secondary: 60K30, 68Q87, 68W40

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 2 • April 2020
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