This paper studies reconstruction of strings based upon their substrings spectrum. Under this paradigm, it is assumed that all substrings of some fixed length are received and the goal is to reconstruct the string. While many existing works assumed that substrings are received error free, we follow in this paper the noisy setup of this problem that was first studied by Gabrys and Milenkovic. The goal of this study is twofold. First we study the setup in which not all substrings in the multispectrum are received, and then we focus on the case where the read substrings are not error free. In each case we provide specific code constructions of strings that their reconstruction is guaranteed even in the presence of failure in either model. We present efficient encoding and decoding maps and analyze the cardinality of the code constructions, while studying the cases where the rates of our codes approach 1.

中文翻译：

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从子串谱重建串

本文研究了基于子弦谱的弦重构。在这种范式下，假设接收到所有固定长度的子串，目标是重建该串。虽然许多现有工作假设子串是无错误接收的，但我们在本文中遵循 Gabrys 和 Milenkovic 首先研究的这个问题的嘈杂设置。这项研究的目标是双重的。首先我们研究不是多谱中的所有子串都被接收到的设置，然后我们关注读取的子串不是没有错误的情况。在每种情况下，我们都提供字符串的特定代码构造，即使在任一模型中出现故障时，也能保证它们的重构。我们提出了有效的编码和解码图，并分析了代码结构的基数，