q元删除信道下的序列重建问题
On Sequence Reconstruction Problem for q-ary Deletion Channels
- Beijing University of Technology（北京工业大学）
- Nankai University（南开大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究q元删除信道下序列重建问题，确定最小Levenshtein距离为2时最大交集N_q(n,2,t)的精确值、极值序列对结构及渐近展开，并证明与q-1情形共享前q-1项。
AI中文摘要：
q元删除信道下的序列重建问题由Levenshtein于2001年提出，关注当每个信道恰好引入t次删除时，唯一恢复传输序列所需的最少信道数。从组合角度看，该问题等价于确定N_q(n,d,t)，即长度为n、字母表为Σ_q={0,1,…,q-1}的q元序列中，两个中心序列的Levenshtein距离至少为d时，其t删除球的最大交集大小。Levenshtein解决了所有n≥t情况下未编码情形N_q(n,1,t)；随后，Gabrys和Yaakobi确定了N_2(n,2,t)，Wang等人将结果推广到N_3(n,2,t)。本文研究最小Levenshtein距离d=2、信道恰好引入t次删除的q元序列问题。我们确定了所有t≥2、q≥4且n足够大时N_q(n,2,t)的精确值，并构造了达到最大交集的显式序列对。此外，对于每个q≥3，我们刻画了所有极值序列对。特别地，若交集大小匹配N_q(n,2,t)的前两项，则两个中心序列必须在相同位置包含形如(a,b,c,a,b)和(b,a,c,b,a)的长度为5的块，其中a,b,c∈Σ_q互不相同；对于t≥q+2，精确最大值N_q(n,2,t)恰好由2q!个具有特定块结构的无序序列对达到。渐近地，我们证明对于q≥4和t≥2，N_q(n,2,t)=6/((t-2)!)n^{t-2}-(3t+13)/((t-3)!)n^{t-3}+(3t^2+25t+64)/(4(t-4)!)n^{t-4}+O(n^{t-5})。此外，N_q(n,2,t)和N_{q-1}(n,2,t)共享前q-1项，且对于t≥q，它们差中n^{t-q}的系数为(6t-6q+5)/((t-q)!)。
英文摘要：
The sequence reconstruction problem for $q$-ary deletion channels, introduced by Levenshtein in 2001, concerns the minimum number of channels required to uniquely recover a transmitted sequence when each channel introduces exactly $t$ deletions. Combinatorially, it is equivalent to determining $N_q(n,d,t)$, the maximum intersection size of two $t$-deletion balls with centers at Levenshtein distance at least $d$, for $q$-ary sequences of length $n$ over the alphabet \(Σ_q=\{0,1,\dots,q-1\}\). Levenshtein solved the uncoded case $N_q(n,1,t)$ for all $n\ge t$; subsequently, Gabrys and Yaakobi determined $N_2(n,2,t)$, and Wang et al. extended the result to $N_3(n,2,t)$. In this paper, we study the problem for \(q\)-ary sequences under minimum Levenshtein distance \(d=2\) with channels that introduce exactly \(t\) deletions. We determine the exact value of \(N_q(n,2,t)\) for all \(t\ge 2, q\geq 4\), and for sufficiently large \(n\), and construct explicit pairs of sequences attaining the maximum intersection. Furthermore, for each $q\ge3$, we characterize all extremal sequence pairs. In particular, if the intersection size matches the first two terms of \(N_q(n,2,t)\), then the two center sequences must contain, at the same positions, length-5 blocks of the forms \((a,b,c,a,b)\) and \((b,a,c,b,a)\) for some distinct \(a,b,c\inΣ_q\); for \(t\ge q+2\), the exact maximum \(N_q(n,2,t)\) is attained precisely by \(2q!\) unordered pairs of sequences with a specific block structure. Asymptotically, we prove that for \(q\ge 4\) and \(t\ge 2\), \[ N_q(n,2,t)=\frac{6}{(t-2)!}n^{t-2}-\frac{3t+13}{(t-3)!}n^{t-3}+\frac{3t^2+25t+64}{4(t-4)!}n^{t-4}+O(n^{t-5}). \] Moreover, \(N_q(n,2,t)\) and \(N_{q-1}(n,2,t)\) share their first \(q-1\) terms, and for \(t\ge q\) the coefficient of \(n^{t-q}\) in their difference is \(\frac{6t-6q+5}{(t-q)!}\).
