The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication setting in which a sender transmits a codeword and the receiver observes K independent noisy versions of this codeword. In this work, we study the problem of efficient reconstruction when each of the $K$ outputs is corrupted by a $q$-ary discrete memoryless symmetric (DMS) substitution channel with substitution probability $p$. Focusing on Reed-Solomon (RS) codes, we adapt the Koetter-Vardy soft-decision decoding algorithm to obtain an efficient reconstruction algorithm. For sufficiently large blocklength and alphabet size, we derive an explicit rate threshold, depending only on $(p, K)$, such that the transmitted codeword can be reconstructed with arbitrarily small probability of error whenever the code rate $R$ lies below this threshold.

翻译：序列重构问题由Levenshtein于2001年提出，其考虑一种通信场景：发送方传输一个码字，而接收方观测到该码字的K个独立噪声版本。本文研究当每个输出均受到替换概率为p的q元离散无记忆对称（DMS）替换信道干扰时的高效重构问题。聚焦于里德-所罗门（RS）码，我们改进Koetter-Vardy软判决译码算法，提出一种高效重构算法。对于足够大的分组长度与字母表规模，我们推导出一个仅依赖于(p, K)的显式速率阈值，使得当码率R低于该阈值时，能以任意小的错误概率重构出发送的码字。