We consider the problem of coding for the substring channel, in which information strings are observed only through their (multisets of) substrings. Due to existing DNA sequencing techniques and applications in DNA-based storage systems, interest in this channel has renewed in recent years. In contrast to existing literature, we consider a noisy channel model where information is subject to noise before its substrings are sampled, motivated by in-vivo storage. We study two separate noise models, substitutions or deletions. In both cases, we examine families of codes which may be utilized for error-correction and present combinatorial bounds on their sizes. Through a generalization of the concept of repeat-free strings, we show that the added required redundancy due to this imperfect observation assumption is sublinear, either when the fraction of errors in the observed substring length is sufficiently small, or when that length is sufficiently long. This suggests that no asymptotic cost in rate is incurred by this channel model in these cases. Moreover, we develop an efficient encoder for such constrained strings in some cases. Finally, we show how a similar encoder can be used to avoid formation of secondary-structures in coded DNA strands, even when accounting for imperfect structures.

翻译：我们研究子串信道的编码问题，其中信息串仅通过其（多重集）子串被观测。由于现有DNA测序技术及其在基于DNA的存储系统中的应用，近年来对该信道的兴趣重新升温。与现有文献不同，我们考虑一种有噪信道模型，其中信息在子串采样前会受到噪声干扰，这源于体内存储的应用场景。我们研究两种独立的噪声模型：替换或删除。在这两种情况下，我们分析可用于纠错的码族，并给出其规模的组合界。通过推广无重复字符串的概念，我们证明：当观测子串长度中的错误比例足够小，或该长度足够长时，由这种不完美观测假设所引入的额外冗余是次线性的。这表明在这些情况下，该信道模型不会产生渐进码率代价。此外，我们在某些情况下为这类受约束字符串设计了高效编码器。最后，我们展示如何利用类似的编码器避免编码DNA链形成二级结构，即使在考虑不完美结构的情况下。