Generalized Unique Reconstruction from Substrings

This paper introduces a new family of reconstruction codes which is motivated by applications in DNA data storage and sequencing. In such applications, strands are sequenced reading some subset their substrings. While previous works considered two extreme cases all substrings pre-defined lengths read or with no overlap for the single string case, this work studies extensions paradigm. The first extension considers setup consecutive given minimum overlap. First, an upper bound provided on attainable rates that guarantee unique reconstruction. Then, efficient constructions asymptotically meet presented. second extension, we study where multiple strings reconstructed together. Given number length, derive lower substrings' length $\ell$ necessary existence multi-strand non-vanishing rates. We then present show approach 1 values behave like bound.