Coding Schemes for the Noisy Torn Paper Channel

TL;DR

This work addresses reconstruction from a noisy torn paper channel modeling DNA strand decay, where substitutions occur with probability $p_s$ and breakage occurs with probability $p_b = \frac{\alpha}{\log_2 n}$. It proposes two marker-based coding schemes: (i) an explicit-indices approach using a 001 marker and de Bruijn-derived indices with an outer LDPC code, and (ii) a data-dependent hashing approach using locality-sensitive hashing (LSH) to replace fixed markers, also concatenated with an outer LDPC code; both employ a beam-search decoder. The results show reconstruction probabilities exceeding $99\%$ in simulations, with static markers performing better at high substitution noise and data-dependent hashing excelling at lower noise, while failures are primarily due to computational resource limits. These schemes advance reliable data recovery for archival DNA storage by combining reassembly constraints with robust error correction and highlight practical open directions including extending to insertions/deletions and experimental validation.

Abstract

To make DNA a suitable medium for archival data storage, it is essential to consider the decay process of the strands observed in DNA storage systems. This paper studies the decay process as a probabilistic noisy torn paper channel (TPC), which first corrupts the bits of the transmitted sequence in a probabilistic manner by substitutions, then breaks the sequence into a set of noisy unordered substrings. The present work devises coding schemes for the noisy TPC by embedding markers in the transmitted sequence. We investigate the use of static markers and markers connected to the data in the form of hash functions. These two tools have also been recently exploited to tackle the noiseless TPC. Simulations show that static markers excel at higher substitution probabilities, while data-dependent markers are superior at lower noise levels. Both approaches achieve reconstruction rates exceeding $99\%$ with no false decodings observed, primarily limited by computational resources.

Figures (8)

• Figure 1: Concatenated coding scheme for the noisy TPC.
• Figure 2: The structure of the coding scheme presented in Section \ref{['sec:marker']} is illustrated in the upper word. Markers, followed by indices and parity bits are interleaved into an LDPC codeword. The colors and the values inside the word and the fragments are referred to in Section \ref{['sec:marker']}. Moreover, affixing the fragments ${\boldsymbol f}_1$ and ${\boldsymbol f}_2$ at indices $4$ and $15$, respectively, constitutes assembly $\mathbf{a}$, which is a partial assembly. Affixing fragments ${\boldsymbol f}_3$ and ${\boldsymbol f}_4$ in the gaps results in the complete assembly $\mathbf{b}$.
• Figure 3: Markers (that are independent of the data) are highlighted above in green. Breakages between markers resulted in fragments ${\boldsymbol f}_2, {\boldsymbol f}_3, {\boldsymbol f}_5$, and ${\boldsymbol f}_7$. As illustrated all possible permutations of these fragments result in a word that satisfies the markers and the indices constraints. Thus, they are all likely to be the correct permutation of the fragments.
• Figure 4: Nested encoding with $L=3$ layers and branching factor $m=2$. Each block contains $d$ data bits and $p_\ell$ parity bits at layer $\ell$.
• Figure 5: Subset selection schemes for Locality-Sensitive Hashing (LSH). Each parity bit is derived from a majority vote over the similarly highlighted data bits.