发表机构
Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了所有列表可解码删除码的最优大小下界，改进已知结果，并给出首个非平凡下界，同时证明公共子序列和超序列数量的上界。
AI 中文摘要
长度为$n$的二进制$k$-删除码是一组二进制字符串，使得如果我们删除一个字符串中的任意$k$位，留下一个长度为$(n-k)$的二进制字符串，我们可以唯一地恢复该码字。在本文中，我们考虑$t$-列表可解码的删除码，其中在码字的$k$位被删除后，我们可以识别一个大小至多为$t$的列表，使得原始码字位于该列表中。我们证明了$t$-列表可解码的$k$-删除码的最优大小的下界为$\Omega_k(2^n t\log^{1/t}n/n^{k+k/t})$，与之前已知的$2$-列表可解码的$2$-删除码的最佳界相比，给出了$\sqrt{\log n}$的改进，并在$t>2$或$k>2$时提供了第一个非平凡的下界。我们的界对所有$t\leq n^k$成立，表明$t=\Omega(\log n)$-列表可解码的删除码具有最优大小$\Theta_k(2^n t/n^k)$，渐近地匹配已知的上界。我们还证明了任意两个二进制字符串的给定长度的公共子序列和公共超序列的数量的上界。
英文摘要
A length-$n$ binary $k$-deletion code is a set of binary strings such that if we delete any $k$ bits of a string, leaving a length-$(n-k)$ binary string, we can uniquely recover the codeword. In this paper, we consider $t$-list decodable deletion codes, where after $k$ bits of a codeword are deleted, we can identify a list of size at most $t$ such that the original codeword lies in the list. We prove a lower bound of $Ω_k(2^n t\log^{1/t}n/n^{k+k/t})$ on the optimal size of a $t$-list decodable $k$-deletion code, giving a $\sqrt{\log n}$ improvement over the previously best known bounds for $2$-list decodable $2$-deletion codes [GH21] and providing the first nontrivial lower bound when $t>2$ or $k>2$. Our bound holds for all $t\leq n^k$, showing that $t=Ω(\log n)-$list decodable deletion codes have optimal size $Θ_k(2^n t/n^k)$, asymptotically matching the known upper bound. We also prove upper bounds on the number of common subsequences and common supersequences of a given length for any two binary strings.
