An Optimal Sequence Reconstruction Algorithm for Reed-Solomon Codes

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a scenario where the sender transmits a codeword from some codebook, and the receiver obtains
$N$
noisy outputs of the codeword. We study the problem of efficient reconstruction using
$N$
outputs that are corrupted by substitutions. Specifically, for the ubiquitous Reed-Solomon codes, we adapt the Koetter-Vardy soft-decoding algorithm, presenting a reconstruction algorithm capable of correcting beyond Johnson radius. Furthermore, the algorithm uses
$\mathrm{O}(nN)$
field operations, where
$n$
is the codeword length.