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 each corrupted by at most $t$ 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 $\mathcal{O}(nN)$ field operations, where $n$ is the codeword length.

翻译：序列重构问题由Levenshtein于2001年提出，该问题考虑以下场景：发送端从某个码本中传输一个码字，接收端获取该码字的$N$个含噪输出。本文研究利用至多被$t$次替换干扰的$N$个输出进行高效重构的问题。具体而言，针对广泛应用的Reed-Solomon码，我们改进了Koetter-Vardy软判决译码算法，提出了一种能够纠正超过Johnson半径的重构算法。此外，该算法使用$\mathcal{O}(nN)$次域操作，其中$n$为码字长度。