In the Levenshtein's sequence reconstruction problem a codeword is transmitted through $N$ channels and in each channel a set of errors is introduced to the transmitted word. In previous works, the restriction that each channel provides a unique output word has been essential. In this work, we assume only that each channel introduces a unique set of errors to the transmitted word and hence some output words can also be identical. As we will discuss, this interpretation is both natural and useful for deletion and insertion errors. We give properties, techniques and (optimal) results for this situation. Quaternary alphabets are relevant due to applications related to DNA-memories. Hence, we introduce an efficient Las Vegas style decoding algorithm for simultaneous insertion, deletion and substitution errors in $q$-ary Hamming spaces for $q \geq 4$.

翻译：在Levenshtein序列重构问题中，码字通过$N$个信道传输，每个信道都会在传输词中引入一组错误。在以往的研究中，每个信道提供唯一输出词的限制条件至关重要。本文中，我们仅假设每个信道对传输词引入唯一的错误集合，因此部分输出词也可能相同。正如我们将要讨论的，这种解释对于删除和插入错误既自然又实用。我们针对这种情况给出了性质、方法及（最优）结果。由于DNA存储器相关应用的需求，四进制字母表具有重要意义。为此，我们针对$q \geq 4$的$q$元汉明空间，提出了一种高效的拉斯维加斯式解码算法，可同时处理插入、删除和替换错误。