  We consider the problem of correcting mass readout errors in information
encoded in binary polymer strings. Our work builds on results for string
reconstruction problems using composition multisets [Acharya et al., 2015] and
the unique string reconstruction framework proposed in [Pattabiraman et al.,
2019]. Binary polymer-based data storage systems [Laure et al., 2016] operate
by designing two molecules of significantly different masses to represent the
symbols $\{{0,1\}}$ and perform readouts through noisy tandem mass
spectrometry. Tandem mass spectrometers fragment the strings to be read into
shorter substrings and only report their masses, often with errors due to
imprecise ionization. Modeling the fragmentation process output in terms of
composition multisets allows for designing asymptotically optimal codes capable
of unique reconstruction and the correction of a single mass error
[Pattabiraman et al., 2019] through the use of derivatives of Catalan paths.
Nevertheless, no solutions for multiple-mass error-corrections are currently
known. Our work addresses this issue by describing the first multiple-error
correction codes that use the polynomial factorization approach for the
Turnpike problem [Skiena et al., 1990] and the related factorization described
in [Acharya et al., 2015]. Adding Reed-Solomon type coding redundancy into the
corresponding polynomials allows for correcting $t$ mass errors in polynomial
time using $t^2\, \log\,k$ redundant bits, where $k$ is the information string
length. The redundancy can be improved to $\log\,k + t$. However, no decoding
algorithm that runs polynomial-time in both $t$ and $n$ for this scheme are
currently known, where $n$ is the length of the coded string.
