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• W3175469882 abstract "We prove that there exists an absolute constant <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$delta >0$ </tex-math></inline-formula> such that any binary code <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$Csubset {0,1}^{N} vphantom {_{int }}$ </tex-math></inline-formula> tolerating <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$(1/2-delta)N$ </tex-math></inline-formula> adversarial deletions must satisfy <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$|C|le 2^{ mathop {mathrm {poly}} log N}$ </tex-math></inline-formula> and thus have rate asymptotically approaching 0. This is the first constant fraction improvement over the trivial bound that codes tolerating <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$N/2$ </tex-math></inline-formula> adversarial deletions must have rate going to 0 asymptotically. Equivalently, we show that there exists absolute constants <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$A$ </tex-math></inline-formula> and <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$delta >0$ </tex-math></inline-formula> such that any set <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$Csubset {0,1}^{N}$ </tex-math></inline-formula> of <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$2^{log ^{A} N}$ </tex-math></inline-formula> binary strings must contain two strings <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$c$ </tex-math></inline-formula> and <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$c'$ </tex-math></inline-formula> whose longest common subsequence has length at least <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$(1/2+delta)N$ </tex-math></inline-formula> . As an immediate corollary, we show that <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$q$ </tex-math></inline-formula> -ary codes tolerating a fraction <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$1-(1+2delta)/q$ </tex-math></inline-formula> of adversarial deletions must also have rate approaching 0. Our techniques include string regularity arguments and a structural lemma that classifies binary strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence." @default.
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• W3175469882 date "2023-04-01" @default.
• W3175469882 modified "2023-10-18" @default.
• W3175469882 title "The Zero-Rate Threshold for Adversarial Bit-Deletions is Less Than 1/2" @default.
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• W3175469882 doi "https://doi.org/10.1109/tit.2022.3223023" @default.
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