Generalized Unique Reconstruction from Substrings

TL;DR

This paper develops new reconstruction codes for DNA sequencing that handle overlapping substrings and multiple strings, providing bounds and constructions that optimize the trade-off between read length and reconstruction accuracy.

Contribution

It introduces two extensions of substring reconstruction codes: one with overlapping reads and another for multiple strings, along with bounds and near-optimal constructions.

Findings

Upper bounds on code rates with overlapping substrings

Constructions that asymptotically meet the bounds

Lower bounds on read length for multi-string reconstruction

Abstract

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