We prove that there exists an absolute constant $\delta>0$ such any binary code $C\subset\{0,1\}^N$ tolerating $(1/2-\delta)N$ adversarial deletions must satisfy $|C|\le 2^{\text{poly}\log N}$ 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.

中文翻译：

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对抗性比特删除的零率阈值小于 1/2

我们证明存在一个绝对常数 $\delta>0$ 这样任何允许 $(1/2-\delta)N$ 对抗性删除的二进制代码 $C\subset\{0,1\}^N$ 必须满足 $| C|\le 2^{\text{poly}\log N}$ 并且因此具有渐进地接近 0 的速率。这是对容忍 $N/2$ 对抗性删除的代码必须具有速率的平凡界限的第一个常数分数改进渐近到 0。等价地，我们证明存在绝对常数 $A$ 和 $\delta>0$，使得 $2^{\log^AN}$ 二进制字符串的任何集合 $C\subset\{0,1\}^N$ 必须包含两个字符串 $c$ 和 $c'$，它们的最长公共子序列的长度至少为 $(1/2+\delta)N$。作为直接推论，我们证明了 $q$-ary 代码可以容忍一小部分 $1-(1+2\delta)/q$ 的对抗性删除，也必须具有接近 0 的比率。我们的技术包括字符串正则性参数和通过振荡模式对二进制字符串进行分类的结构引理。利用这些工具，我们可以在任何大型代码中找到两个具有相似振荡模式的字符串，利用它们来找到一个长的公共子序列。