发表机构
Tel-Hai University; MIGAL – Galilee Research Institute（泰尔海大学; 米加尔-加利利研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究纠正多重反向互补与回文重复操作的q元码，提出逐坐标双射实现两类错误纠正码的转换，构造图并经贪心着色得到冗余度为4 log_q n + O_{q,k}(1)的存在性码，二元两次错误问题的存在性界存在因子2的差距。
AI 中文摘要
反向互补（RC）重复与回文（PAL）重复会复制长度为k的块，将副本反转后插入原块旁；RC重复还会对复制的符号进行互补操作。我们研究能纠正t次此类连续操作的q元码，后续操作可能复制先前操作生成的符号。对于固定的q、k、t，任一信道的长度为n的码C满足n - log_q|C| ≥ t log_q n - O_{q,k,t}(1)；当q、k固定且t = o(n)时，下界为t log_q(n/t) - O_{q,k}(t)。对于偶字母集上的单个RC错误（带有无不动点的互补操作），此前已知的RC专用提升方法仅适用于奇数k，未覆盖偶数k。对于所有偶数k，我们提出一种逐坐标双射，可将每个RC重复转化为PAL重复。将此双射应用于每个码字，能将任意t次错误纠正的RC码转换为相同大小的PAL码，反之亦然；编码器与解码器可通过添加线性时间的坐标遍历实现转换。我们还确定了从一个源字经恰好两次错误操作产生的不同后代的最大数量。对于PAL，在任意两个不同字母符号间交替的字可达到此最大值，而该双射给出了RC的最大化者。对于偶数k的PAL和RC，我们构造一个图，其顶点为所有长度为n的q元字，当且仅当两个不同字有共同的精确两次错误后代时，将二者相连。通过界定度数并进行贪心着色，可得到冗余度为4 log_q n + O_{q,k}(1)的存在性码。在二元两次错误问题中，其逆给出2 log_2 n - O_k(1)，使最佳存在性界存在因子2的差距。
英文摘要
Reverse-complement (RC) and palindromic (PAL) duplications copy a length-$k$ block, reverse the copy, and insert it immediately after the original block; an RC duplication also complements the copied symbols. We study $q$-ary codes correcting $t$ such operations performed sequentially, so a later operation may copy symbols created by an earlier one. For fixed $q\geq2$ and $k,t\geq1$, every length-$n$ code $C$ for either channel satisfies $n-\log_q|C|\geq t\log_q n-O_{q,k,t}(1)$; for fixed $q,k$ and $1\leq t=t(n)=o(n)$ the lower bound is $t\log_q(n/t)-O_{q,k}(t)$. For a single RC error over an even alphabet with a fixed-point-free complement, the previously known RC-specific construction applies at odd $k$ and does not cover even $k$. For every even $k$ and any involutive complement, we give a coordinate-wise bijection that turns each RC duplication into a PAL duplication. Applying this bijection to every codeword therefore converts any $t$-error-correcting RC code into a PAL code of the same size, and conversely; encoders and decoders transfer by adding linear-time coordinate passes. For every even $k$, we also determine the maximum number of distinct descendants produced by exactly two errors from one source word. Words alternating between any two distinct alphabet symbols attain this maximum for PAL, and the bijection gives the RC maximizers. For both PAL and RC at fixed even $k$, we prove the existence of two-error-correcting codes with redundancy $4\log_q n+O_{q,k}(1)$. In the binary two-error problem, the converse gives $2\log_2 n-O_k(1)$, leaving a factor-two gap in the best existence bounds.
Comments33 pages. Submitted to Designs, Codes and Cryptography. Exposition, attribution and proof explanations clarified; results unchanged
