关于回文/反向互补重复纠错码
On the Palindromic/Reverse-Complement Duplication Correcting Codes
AI总结：
本文提出两种显式构造的纠错码，能够纠正长度为1的反向互补重复，其中一种冗余度更低且复杂度更优。
摘要：
受体内DNA存储应用的启发，我们研究了纠正重复的编码方法。一个反向互补重复是将长度为k的子串的反转和互补副本插入到其原始位置旁边，而回文重复仅插入反转副本而不进行互补。我们首先构造了一个具有单个冗余符号的显式码，能够纠正任意数量的反向互补重复（相应地，回文重复），前提是所有重复的长度k≥3⌈log_q n⌉且互不重叠。接着，我们推导出能够纠正任意长度反向互补重复（相应地，回文重复）的Gilbert-Varshamov界，证明最优冗余被上界限制为2log_q n + log_q log_q n + O(1)。最后，对于q≥4，我们提出两种显式构造的码，能够纠正t个长度为1的反向互补重复。第一种构造实现了冗余度为2tlog_q n + O(log_q log_q n)，编码复杂度为O(n)，解码复杂度为O(n(log_2 n)^4)。第二种构造实现了改进的冗余度为(2t-1)log_q n + O(log_q log_q n)，但编码和解码复杂度为O(n·poly(log_2 n))。
英文摘要：
Motivated by applications in in-vivo DNA storage, we study codes for correcting duplications. A reverse-complement duplication of length $k$ is the insertion of the reversed and complemented copy of a substring of length $k$ adjacent to its original position, while a palindromic duplication only inserts the reversed copy without complementation. We first construct an explicit code with a single redundant symbol capable of correcting an arbitrary number of reverse-complement duplications (respectively, palindromic duplications), provided that all duplications have length $k \ge 3\lceil \log_q n \rceil$ and are disjoint. Next, we derive a Gilbert-Varshamov bound for codes that can correct a reverse-complement duplication (respectively, palindromic duplication) of arbitrary length, showing that the optimal redundancy is upper bounded by $2\log_q n + \log_q\log_q n + O(1)$. Finally, for $q \ge 4$, we present two explicit constructions of codes that can correct $t$ length-one reverse-complement duplications. The first construction achieves a redundancy of $2t\log_q n + O(\log_q\log_q n)$ with encoding complexity $O(n)$ and decoding complexity $O\big(n(\log_2 n)^4\big)$. The second construction achieves an improved redundancy of $(2t-1)\log_q n + O(\log_q\log_q n)$, but with encoding and decoding complexities of $O\big(n \cdot \mathrm{poly}(\log_2 n)\big)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
