The zero-rate threshold for adversarial bit-deletions is less than 1/2

We prove that there exists an absolute constant 6 > 0 such any binary code
$C$
⊂ {0, 1}
^N
tolerating (1/2 - δ)
$N$
adversarial deletions must satisfy| C| ≤ 2
^polylog
$N$
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 6 > 0 such that any set
$C$
⊂ {0, 1} of 2
^log
A
N binary strings must contain two strings
$c$
and c’ whose longest common subsequence has length at least (1/2 + δ) N. As an immediate corollary, we show that q-ary codes tolerating a fraction 1 - (1 + 2δ) /
$q$
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.