Coding for Sequence Reconstruction for Single Edits

Abstract

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. The common setup assumes the codebook to be the entire space and the problem is to determine the minimum number of distinct reads that is required to reconstruct the transmitted codeword.Motivated by modern storage devices, we study a variant of the problem where the number of noisy reads N is fixed. Specifically, we design reconstruction codes that reconstruct a codeword from N distinct noisy reads. We focus on channels that introduce single edit error (i.e. a single substitution, insertion, or deletion) and their variants, and design reconstruction codes for all values of N . In particular, for the case of a single edit, we show that as the number of noisy reads increases, the number of redundant bits required can be gracefully reduced from log n + O(1) to log log n + O(1), and then to O(1), where n denotes the length of a codeword. We also show that the redundancy of certain reconstruction codes is within one bit of optimality.

The Error Probability of Maximum-Likelihood Decoding over Two Deletion/Insertion Channels

“…The most relevant case of the reconstruction problem to our work is the one studied in [7], where it was shown how the shifted VT codes can be used for the two single-deletion channels case. In our parallel work [22] the dual problem is studied where the number of channels is given and then the goal is to find the best code which guarantees successful decoding in the worst case. Hence, the problem studied in this paper can be regarded as the probabilistic variant of the dual problem of the reconstruction problem.…”

“…[35], [36], where reconstruction algorithms for the maximum-likelihood have been studied. Abroshan et al presented in [1] a new coding scheme for sequence reconstruction which is based on the Varshamov Tenegolts (VT) code [40] and in a parallel work [22] it is studied how to design codes for the worst case, when the number of channels is given.…”

Trace Reconstruction Problems in Computational Biology

“…In the larger redundancy regime, such as redundancy εn with ε ∈ (0, 1) being a constant, an improved trace complexity of exp(O(log 1/3 (1/ε))) is achievable [6]. Recent work also more thoroughly studies coded trace reconstruction in the insertion/deletion channel when there are a constant number of errors or a constant number of traces [5], [92], [108]- [110]. Before integrating these results into a DNA data storage system, certain ulterior constraints should be addressed as well.…”

Sequences and their applications

“…Motivated by modern storage devices, Kiah et al [60] introduced a variant of the sequence reconstruction problem where the number of noisy reads N is fixed. Hence, our fundamental problem is then: how large can this codebook C be?…”

“…Sequence Reconstruction for Single-deletion channel : Modifying a code construction in [58], Kiah et al proposed in [60] a reconstruction code for the singledeletion channel for N = 2 with log 2 log 2 n+O(1) bits of redundancy. For this case, we focus on the converse of the problem and show that log 2 log 2 n−O(1) redundant bits are necessary, and therehby demonstrating that the code construction provided in [60] is asymptotically optimal. Furthermore, we show that these reconstruction codes can be used in t-deletion channels (with t ≥ 2) to uniquely reconstruct codewords from n t−1 + O (n t−2 ) distinct noisy reads.…”