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Geno-Weaving: Low-Complexity Capacity-Achieving DNA Storage

Hsin-Po Wang, Venkatesan Guruswami

TL;DR

This paper lays down a rateless code along each strand to encode its index; it then lays down a capacity-achieving block code at the same position across all strands to protect data, and weaves a low-complexity coding scheme that achieves DNA's capacity.

Abstract

As a possible implementation of data storage using DNA, multiple strands of DNA are stored in a liquid container so that, in the future, they can be read by an array of DNA readers in parallel. These readers will sample the strands with replacement to produce a random number of noisy reads for each strand. An essential component of such a data storage system is how to reconstruct data out of these unsorted, repetitive, and noisy reads. It is known that if a single read can be modeled by a substitution channel $W$, then the overall capacity can be expressed by the "Poisson-ization" of $W$. In this paper, we lay down a rateless code along each strand to encode its index; we then lay down a capacity-achieving block code at the same position across all strands to protect data. That weaves a low-complexity coding scheme that achieves DNA's capacity.

Geno-Weaving: Low-Complexity Capacity-Achieving DNA Storage

TL;DR

This paper lays down a rateless code along each strand to encode its index; it then lays down a capacity-achieving block code at the same position across all strands to protect data, and weaves a low-complexity coding scheme that achieves DNA's capacity.

Abstract

As a possible implementation of data storage using DNA, multiple strands of DNA are stored in a liquid container so that, in the future, they can be read by an array of DNA readers in parallel. These readers will sample the strands with replacement to produce a random number of noisy reads for each strand. An essential component of such a data storage system is how to reconstruct data out of these unsorted, repetitive, and noisy reads. It is known that if a single read can be modeled by a substitution channel , then the overall capacity can be expressed by the "Poisson-ization" of . In this paper, we lay down a rateless code along each strand to encode its index; we then lay down a capacity-achieving block code at the same position across all strands to protect data. That weaves a low-complexity coding scheme that achieves DNA's capacity.
Paper Structure (17 sections, 6 theorems, 43 equations, 5 figures)

This paper contains 17 sections, 6 theorems, 43 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical model of DNA coding
  3. Understanding the capacity
  4. Capacity Bound of Sampling
  5. Decomposition of the mutual information
  6. Technical details regarding Poisson limit theorem
  7. DNA Coding by Capacity-Achieving Codes
  8. Code selection (technical and skippable)
  9. Encoding and code rate
  10. Decoding and error probability
  11. Capacity Bound of Shuffling
  12. Geno-Weaving with Rateless and Block Codes
  13. A straw man with some unrealistic assumptions
  14. Encoding and code rates of the straw man
  15. Decoding of the straw man
  16. ...and 2 more sections

Key Result

Theorem 4

SupposeThis is equivalent to assuming that each $X^s$ spends $\log_q n$ letters of their length $\ell + \log_q n$ to encode the strand index $s \in [n]$, and the indexing part is error-free. that a genie reveals $S$. Suppose that, as $\ell$, $m$, and $n$ increase, $\lambda \coloneqq m/n$ remains a c That is, the capacity of DNA coding is determined by the Poisson-ization of the channel $W$ that mo

Figures (5)

  • Figure 1: DNA strands float in liquid; nanopores will read them in parallel. See WZB21 for a more accurate picture.
  • Figure 2: To prove Theorem \ref{['thm:useC']}, we will apply a block code at the same position across all strands to protect data.
  • Figure 3: To prove Theorem \ref{['thm:useBR']}, we will deploy a rateless code on each strand to encode its index and a block code at each position to protect data. The code rate of the latter will depend on $p$. The letter at any intersection will be the mod-$4$ sum of the rateless code symbol and the block code symbol.
  • Figure 4: The decoding part of Theorem \ref{['thm:useBR']}. Step 1: decode $\mathcal{R}$ to obtain indices. Even number steps: subtract indices from $X$ and decode $\mathcal{B}$ to obtain data. Odd number steps: subtract data from $X$ and decode $\mathcal{R}$ to obtain indices.
  • Figure 5: Step 1: decode indices. Even number steps: subtract indices to decode data. Odd number steps: subtract data to decode indices. This figure caps the number of reads at $\kappa \coloneqq 9$.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 4
  • Theorem 5: presented in ITA 2024
  • Theorem 6: ShH21
  • Theorem 7: presented in ISIT 2024's satellite workshop
  • Proposition 8
  • Lemma 9