AI 中文总结
研究有限置换信道的纠错码，引入消光深度概念，通过后续块扩展跟踪未解决块对，给出q = 3和q = 5的块集及相关理论，开发检测理论并扩展准则到固定位移半径r，改进了一些下界。
AI 中文摘要
在半径为1的有限置换信道中，错误由不相交的相邻换位组成。纠错码必须区分不同的码字：它们的错误球不能包含共同的接收字。汉明距离不能确保这一点，因为不相交的交换会使在许多位置不同的字变得可混淆。对于分组级联码，早期工作仅通过较长的初始块测试每个可能的冲突。这个充分条件不是必要的：我们展示了一个有效的三元块集，它不能通过该测试。我们引入了消光深度，它通过后续块扩展跟踪未解决的第一块对，并证明它们在一个共同的有限视界处的消光证明了在每个长度处的纠错。该准则给出了明确的q = 3和q = 5的块集，速率分别高于0.6777475和0.6694926。不存在与块集无关的深度界：我们给出了一个精确的线性族，验证了3≤k≤60的二次公式，并推导了多项式时间有限图测试。对于不断增长的字母表，归一化纠错损失介于lnφ和ln(1.82560995)之间，而明确的有限长度覆盖改进了有限字母表的上界。我们开发了一种用于检测的并行定向消光理论，包括一个全长度块准则、一个早期测试未解决的集以及精确的走廊深度4k。稳定类型提升产生最优的一阶损失√2，而弱之字形码改进了q = 3,4的下界。最后，窗口限制、抵消和有界待处理输入前沿将该准则及其有限状态验证扩展到每个固定位移半径r。
英文摘要
In the radius-one limited permutation channel, errors consist of disjoint adjacent transpositions. A correcting code must separate distinct codewords: their error balls may not contain a common received word. Hamming distance does not ensure this, because disjoint swaps can make words differing in many positions confusable. For block-concatenation codes, earlier work tested each possible collision only through the longer initial block. This sufficient condition is not necessary: we exhibit a valid ternary block set that it does not certify. We introduce extinction depth, which tracks unresolved first-block pairs through later block extensions, and prove that their extinction at one common finite horizon certifies correction at every length. The criterion gives explicit $q=3$ and $q=5$ block sets with rates above $0.6777475$ and $0.6694926$. No block-set-independent depth bound exists: we give an exact linear family, verify a quadratic formula for every $3\leq k\leq 60$, and derive a polynomial-time finite-graph test. For growing alphabets, the normalized correction loss lies between $\lnφ$ and $\ln(1.82560995)$, while explicit finite-length covers improve finite-alphabet upper bounds. We develop a parallel directed-extinction theory for detection, including an all-length block criterion, a set unresolved by the earlier test, and exact corridor depth $4k$. Stable type lifting yields optimal first-order loss $\sqrt{2}$, and weak-zigzag codes improve the $q=3,4$ lower bounds. Finally, window restriction, cancellation, and a bounded pending-input frontier extend the criterion and its finite-state verification to every fixed displacement radius $r$.
Comments44 pages, 6 tables. Accompanying reproducibility package: https://doi.org/10.5281/zenodo.21398315
