[Paper Review] Coding Schemes for the Noisy Torn Paper Channel

Frederik Walter, Maria Abu-Sini+2 more|arXiv (Cornell University)|Jan 16, 2026

DNA and Biological Computing0 citations

TL;DR

The paper develops two coding schemes for the noisy torn paper channel: a marker-based scheme with explicit indexing and a hash-based scheme using locality-sensitive hashing, both paired with an outer LDPC code, achieving over 99% reconstruction with no false decodings in simulations.

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.

Motivation & Objective

• Motivate long-term DNA data storage by modeling decay as a probabilistic torn paper channel with substitutions and breakages.
• Design coding schemes that enable reconstruction from unordered, noisy fragments.
• Assess the trade-offs between static markers and data-dependent hashing under varying noise levels.
• Evaluate reconstruction performance and practical decoding complexity through simulations.

Proposed method

• Two concatenated coding schemes are proposed: (1) marker-based with explicit indexing using markers, de Bruijn sequence indices, and parity linking blocks; (2) hash-based without explicit indices using locality-sensitive hashing to preserve fragment connectivity under substitutions.
• An outer LDPC code corrects substitutions and ties the reassembly to error correction.
• Marker scheme uses fixed 001 markers plus repeated de Bruijn bits to robustly detect indices in noisy fragments.
• Data-dependent scheme employs LSH to maintain fragment adjacency despite substitutions, with robust parity-distance matching during reassembly.
• A beam-search reconstruction algorithm decodes partial assemblies by minimizing constraint violations before applying LDPC belief propagation.
• Simulation parameters explore breakage probability α and substitution p_s, evaluating achievable rates against channel capacity.

Figure 1 : Concatenated coding scheme for the noisy TPC.

Experimental results

Research questions

• RQ1 How can coding schemes enable reliable reassembly of fragmented, noisy DNA-derived sequences under substitution and breakage noise?
• RQ2 What are the comparative benefits of static markers versus data-dependent hashing for reassembly and error correction in the torn paper channel?
• RQ3 Can concatenated coding with LDPC outer codes achieve high reconstruction rates while managing computational complexity?
• RQ4 How do index-based marker schemes and LSH-based hashing perform across noise regimes and fragmentations?
• RQ5 What are the practical parameter regimes (d, d', c1, c2, etc.) that maximize rate while ensuring reliable reconstruction?

Key findings

• Both marker-based and hash-based schemes achieve reconstruction probabilities above 99% in simulations.
• Static markers outperform data-dependent hashing at higher substitution probabilities, while data-dependent hashing (LSH) performs better at lower noise levels.
• No false decodings were observed in simulations; failures were due to computational resource limits or search-time constraints rather than incorrect assemblies.
• Marker-based indexing with explicit indices shows favorable rate performance in regimes with higher breakage, while the LSH-based approach yields robustness against substitutions.
• Outer LDPC codes provide error correction, with decoding performed via min-sum belief propagation after assembly.

Figure 2 : The structure of the coding scheme presented in Section II 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 II . Moreover, affixing t