Motivated by average-case trace reconstruction and coding for portable DNA-based storage systems, we initiate the study of , the design and analysis of high-rate efficiently encodable codes that can be efficiently decoded with high probability from few reads (also called ) corrupted by edit errors. Codes used in current portable DNA-based storage systems with nanopore sequencers are largely based on heuristics, and have no provable robustness or performance guarantees even for an error model with i.i.d. deletions and constant deletion probability. Our work is a first step towards the design of efficient codes with provable guarantees for such systems. We consider a constant rate of i.i.d. deletions, and perform an analysis of marker-based code-constructions. This gives rise to codes with redundancy @math (resp. @math ) that can be efficiently reconstructed from @math (resp. @math ) traces, where @math is the message length. Then, we give a construction of a code with @math bits of redundancy that can be efficiently reconstructed from @math traces if the deletion probability is small enough. Finally, we show how to combine both approaches, giving rise to an efficient code with @math bits of redundancy which can be reconstructed from @math traces for a small constant deletion probability.
Coded trace reconstruction is also related to the multi-use deletion channel. In fact, the decoding problem for this channel can be interpreted as coded trace reconstruction with a fixed number of traces. Some results are known about the capacity of this channel for small deletion probability @cite_49 , and about maximum likelihood decoding @cite_31 .
@cite_31 @cite_49
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This paper studies problems in data reconstruction, an important area with numerous applications. In particular, we examine the reconstruction of binary and nonbinary sequences from synchronization (insertion deletion-correcting) codes. These sequences have been corrupted by a fixed number of symbol insertions (larger than the minimum edit distance of the code), yielding a number of distinct traces to be used for reconstruction. We wish to know the minimum number of traces needed for exact reconstruction. This is a general version of a problem tackled by Levenshtein for uncoded sequences. We introduce an exact formula for the maximum number of common supersequences shared by sequences at a certain edit distance, yielding an upper bound on the number of distinct traces necessary to guarantee exact reconstruction. Without specific knowledge of the code words, this upper bound is tight. We apply our results to the famous single deletion insertion-correcting Varshamov–Tenengolts (VT) codes and show that a significant number of VT code word pairs achieve the worst case number of outputs needed for exact reconstruction. We also consider extensions to other channels, such as adversarial deletion and insertion deletion channels and probabilistic channels. Motivated by the error behavior in the DNA storage channel, in this paper, we extend the previously studied sequence reconstruction problem by Levenshtein. The reconstruction problem studies the model in which the information is read through multiple noisy channels, and the decoder, which receives all channel estimations, is required to decode the information. For the combinatorial setup, the assumption is that all the channels cause at most some @math errors. Levenshtein considered the case in which all the channels have the same behavior, and we generalize this model and assume that the channels are not identical. Thus, different channels may cause different maximum numbers of errors. For example, we assume that there are @math channels, which cause at most @math or @math errors, where @math t_ 1 @math pN @math 0 . If the information codeword belongs to a code with minimum distance @math , the problem is then to find the minimum number of channels that guarantees successful decoding in the worst case. A different problem we study in this paper is where the number of channels is fixed, and the question is finding the minimum distance @math that provides exact reconstruction. We study these problems and show how to apply them for the cases of substitutions and transpositions.