AI 中文总结
研究抗破坏码，针对长度为\(n\)且最多\(t\)次破坏的二进制码字，在\(t\leq n^{1 - \varepsilon}\)时，提出计算消息短代数指纹的方法，构造出冗余为\(O(t\log n)\)的抗破坏码，弥合了已知构造与信息论下界的差距。
AI 中文摘要
抗破坏码可保护一个字免受全知对手在相邻符号间任意边界处的破坏。对于长度为\(n\)且最多有\(t\)次破坏的二进制码字，该模型最著名的显式构造冗余为\(O(t\log n\log\log\log n)\)，而信息论下界为\(\Omega(t\log (n/t))\)。本文通过给出一个当\(t\leq n^{1 - \varepsilon}\)（\(\varepsilon\in(0,1)\)固定）时冗余为\(O(t\log n)\)的抗破坏码来弥合这一差距，使其与信息论下界在常数因子范围内匹配。关键思想是计算消息的短代数指纹，使解码器能拒绝接收到的片段的错误组合。
英文摘要
Break-resilient codes enable reliable communication in the presence of an omniscient adversary that may split a transmitted message at arbitrary boundaries between consecutive symbols, while the receiver observes only an unordered multiset of the resulting fragments. For binary codewords of length~$n$ subject to at most~$t$ breaks, the best known explicit construction has redundancy~$O(t\log_2 n\log_2\log_2\log_2 n)$, whereas the information-theoretic lower bound is~$Ω(t\log_2(n/t))$. In this paper, we extend the binary break model to any fixed finite field~$\bbF_q$ and establish a redundancy lower bound of~$Ω(t\log_q(n/t))$. We then give an explicit construction of~$q$-ary break-resilient codes with redundancy~$O(t\log_q n)$ when~$t\leq n^{1-\varepsilon}$ for a fixed constant~$\varepsilon\in(0,1)$, matching the information-theoretic lower bound up to a constant factor. The key idea is to compute a short algebraic fingerprint of the message, which enables the decoder to reject incorrect assemblies of the received fragments.
