On Sequence Reconstruction Problem for q-ary Deletion Channels

RESUMO

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 \(\Sigma_q=\{0,1,\dots,q-1\}\). 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 \(a,b,c\in\Sigma_q\); 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 \(\frac{6t-6q+5}{(t-q)!}\).