Coding for Sequence Reconstruction for Single Edits

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. The common setup assumes the codebook to be the entire space and the problem is to determine the minimum number of distinct reads that is required to reconstruct the transmitted codeword. Motivated by modern storage devices, we study a variant of the problem where the number of noisy reads
$N$
is fixed. Specifically, we design
reconstruction codes
that reconstruct a codeword from
$N$
distinct noisy reads. We focus on channels that introduce a single edit error (i.e. a single substitution, insertion, or deletion) and their variants, and design reconstruction codes for all values of
$N$
. In particular, for the case of a single edit, we show that as the number of noisy reads increases, the number of redundant symbols required can be gracefully reduced from
$\log _{q} n+O(1)$
to
$\log _{q} \log _{q} n+O(1)$
, and then to
$O(1)$
, where
$n$
denotes the length of a codeword. We also show that these reconstruction codes are asymptotically optimal. Finally, via computer simulations, we demonstrate that in certain cases, reconstruction codes can achieve similar performance as classical error-correcting codes with less redundant symbols.