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.

翻译：我们证明存在绝对的常数$delta>0 这样的二进制代码 $C\ subset=0,1 ⁇ N$(1/2-delta)N$(1/2-delta)N$(1/2_Text{poly ⁇ logN}$),因此,其速率总是接近0。这是比允许$N/2的对称删除的代码的速率必须达到零的微小约束值的第一个常数改进。同样,我们显示存在绝对的常数$和$(1/2-delta)0美元($)N$(1/2-delta)N$(美元)N美元),因此任何设定的对称的对称删除必须满足$C\ subset=10,1 ⁇ N$($)N$($)N}$($)必须包含两个字符串($和$美元),其最长的子序列长度长度长度至少为$(1/2 ⁇ )N。作为直接的必然结果,我们显示, $-rideal co codeal codeal code develilate develilationslationslationslation (我们通过这些技术可以找到两个共同的参数) 。