发表机构
Institute of Mathematics and Interdisciplinary Sciences, Xidian University; School of Mathematical Sciences, Capital Normal University（西安电子科技大学数学与 interdisciplinary 科学研究院; 首都师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种新定位方法，构造了$q$元$k$-子串编辑纠正码，冗余为$\log n+8\log\log n+o(\log\log n)$比特，并进一步构造了冗余$\log n+O_{q,k}(1)$的最优码，首次在固定参数全范围内实现突发删除和局部删除纠正码的最优冗余。
AI 中文摘要
序列中的$k$-子串编辑首先删除长度至多为$k$的子串，然后在同一位置插入长度至多为$k$的序列。能够纠正$k$-子串编辑的码称为$k$-子串编辑码。本文开发了一种新的定位方法，并利用它构造了$q$元$k$-子串编辑纠正码，对于任意固定的$q\ge2$和$k\ge1$，其冗余为$\log n+8\log\log n+o(\log\log n)$比特，其中$n$为码长。对于二进制字母表，这改进了Li等人获得的冗余$\log n+16k\log\log n+o(\log\log n)$。当删除的子串与插入的序列长度不同时，我们进一步构造了冗余为$\log n+O_{q,k}(1)$的码，这在加性常数意义下是最优的。作为推论，对于所有固定的$q\ge2$和$1\le t\le T$，我们分别获得了冗余为$\log n+O_{q,t}(1)$和$\log n+O_{q,T}(1)$的$q$元$(\le t)$-突发删除纠正码和$(t,T)$-局部删除纠正码。据我们所知，这是首次在固定参数全范围内为这两种删除模型实现加性常数意义下最优冗余的构造。对于$t\ge2$的$(\le t)$-突发删除纠正，此前仅在$q=t=2$时由Levenshtein于1967年实现该冗余。
英文摘要
A $k$-substring edit in a sequence first deletes a substring of length at most $k$, and then inserts a sequence of length at most $k$ at the same position. A code that can correct a $k$-substring edit is called a $k$-substring edit code. In this paper, we develop a new localization method and use it to construct a $q$-ary $k$-substring edit correcting code with $\log n+8\log\log n+o(\log\log n)$ bits of redundancy for any fixed $q\ge2$ and $k\ge1$, where $n$ is the code length. For the binary alphabet, this improves upon the redundancy $\log n+16k\log\log n+o(\log\log n)$ obtained by Li \emph{et al}. When the deleted substring and the inserted sequence have different lengths, we further construct codes with redundancy $\log n+O_{q,k}(1)$, which is optimal up to an additive constant. As corollaries, for all fixed $q\ge2$ and $1\le t\le T$, we obtain $q$-ary $(\le t)$-burst-deletion correcting codes and $(t,T)$-localized deletion correcting codes with redundancies $\log n+O_{q,t}(1)$ and $\log n+O_{q,T}(1)$, respectively. To the best of our knowledge, these are the first constructions attaining optimal redundancy up to an additive constant for these two deletion models over the full range of fixed parameters. For $(\le t)$-burst-deletion correction with $t\ge2$, such redundancy had previously been achieved only for $q=t=2$ by Levenshtein in 1967.
Comments35 pages
