Optimal Reconstruction Codes for Deletion Channels

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. Motivated by modern storage devices, we introduced a variant of the problem where the number of noisy reads N is fixed (Kiah et al. 2020). Of significance, for the single-deletion channel, using log
₂
log
₂
n+O(1) redundant bits, we designed a codebook of length n that reconstructs codewords from two distinct noisy reads.In this work, we show that log
₂
log
₂
n-O(1) redundant bits are necessary for such reconstruction codes, thereby, demonstrating the optimality of our previous construction. Furthermore, we show that these reconstruction codes can be used in t-deletion channels (with t ≥ 2) to uniquely reconstruct codewords from n
^t-1
+ O(n
^t-2
) distinct noisy reads.