Sequence Reconstruction under Channels with Multiple Bursts of Insertions or Deletions - Lan, Zhaojun; Sun, Yubo; Yu, Wenjun; Ge, Gennian (IEEE transactions on information theory, 2025)
The sequence reconstruction problem involves a model where a sequence is transmitted over several identical channels. This model investigates the minimum number of channels required for the unique reconstruction of the transmitted sequence. Levenshtein established that this number exceeds the maximum size of the intersection between the error balls of any two distinct transmitted sequences by one. In this paper, we consider channels subject to multiple bursts of insertions and multiple bursts of deletions, respectively, where each burst has an exact length of value b . Our key findings are as follows: * Insertion Case: We investigate b -burst-insertion balls of radius t centered at q -ary sequences of length n . We establish that the size of an error ball is independent of its chosen center. Furthermore, we demonstrate that the intersection between error balls centered at two sequences differing only at their first position yields the largest intersection size, denoted by N q,b + ( n , t ). We also propose a reconstruction algorithm with linear runtime complexity, which processes N q,b + ( n , t )+1 distinct output sequences from the channel to recover the correct transmitted sequence. * Deletion Case: We examine b -burst-deletion balls of radius t centered at q -ary sequences of length n . In contrast to burst-insertion balls, we prove that the size of a burst-deletion ball is dependent on its chosen center. Particularly, we show that the b -burst-deletion ball centered at the b -cyclic sequence 0 b ⸰ 1 b ⸰ · · · ⸰ ( q − 1) b ⸰ 0 b · · · achieves the largest size. For binary alphabets, we then demonstrate that the intersection of b -burst-deletion balls centered at 0 b ⸰ 1 b ⸰ 0 b ⸰ 1 b · · · and 0 b −1 ⸰ 1 ⸰ 1 b ⸰ 0 b ⸰ 1 b · · · yields the largest size, denoted by N 2, b − ( n , t ). Moreover, we propose a reconstruction algorithm with linear runtime complexity, which processes N ≥ N 2, b − ( n , t )+1 distinct output sequences from the channel to reconstruct the correct transmitted sequence 1 .