The Zero-Rate Threshold for Adversarial Bit-Deletions is Less Than 1/2

We prove that there exists an absolute constant
$\delta >0$
such that any binary code
$C\subset \{0,1\}^{N} \vphantom {_{\int }}$
tolerating
$(1/2-\delta)N$
adversarial deletions must satisfy
$|C|\le 2^{ \mathop {\mathrm {poly}} \log N}$
and thus have rate asymptotically approaching 0. This is the first constant fraction improvement over the trivial bound that codes tolerating
$N/2$
adversarial deletions must have rate going to 0 asymptotically. Equivalently, we show that there exists absolute constants
$A$
and
$\delta >0$
such that any set
$C\subset \{0,1\}^{N}$
of
$2^{\log ^{A} N}$
binary strings must contain two strings
$c$
and
$c'$
whose longest common subsequence has length at least
$(1/2+\delta)N$
. As an immediate corollary, we show that
$q$
-ary codes tolerating a fraction
$1-(1+2\delta)/q$
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.