The zero-rate threshold for adversarial bit-deletions is less than 1/2
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We prove that there exists an absolute constant δ>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 δ>0 such that any set C⊂{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+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.
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