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

Abstract

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 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 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.

Reed Solomon Codes Against Adversarial Insertions and Deletions

“…In a recent work, Guruswami, He, and, Li [GHL21] strengthened this result for binary codes and showed that there exists an absolute constant δ such that any binary code C ⊆ {0, 1} n (not only linear) that can decode from (1/2 − δ) fraction of deletions must satisfy |C| ≤ 2 poly log n . In particular, we cannot hope to decode a fraction of insdel errors arbitrarily close to 1/2 with codes of positive rate.…”