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Optimizing the Decoding Probability and Coverage Ratio of Composite DNA

Tomer Cohen, Eitan Yaakobi

TL;DR

This work models composite DNA storage where each symbol encodes a distribution over nucleotides and analyzes two central problems: the coverage depth (how many reads are needed) to decode composite strands, and the design of composite alphabets that maximize decoding success under maximum likelihood decoding. It introduces the composite DNA channel, formalizes observed distributions, and derives exact and asymptotic expressions for the coverage depth E(ℓ, ω), including a closed form for ω=2 and a general tail-sum representation for arbitrary ω. The authors then characterize optimal composite alphabets, showing that including the base alphabet and a uniform mixture often yields optimal or near-optimal decoding performance, with explicit results for the case m=q+1 and a detailed binary alphabet treatment. Overall, the paper provides fundamental bounds, exact results for key special cases, and practical guidance for alphabet design in composite DNA storage, highlighting the trade-offs between average and worst-case decoding performance under ML decoding.

Abstract

This paper studies two problems that are motivated by the novel recent approach of composite DNA that takes advantage of the DNA synthesis property which generates a huge number of copies for every synthesized strand. Under this paradigm, every composite symbols does not store a single nucleotide but a mixture of the four DNA nucleotides. The first problem studies the expected number of strand reads in order to decode a composite strand or a group of composite strands. In the second problem, our goal is study how to carefully choose a fixed number of mixtures of the DNA nucleotides such that the decoding probability by the maximum likelihood decoder is maximized.

Optimizing the Decoding Probability and Coverage Ratio of Composite DNA

TL;DR

This work models composite DNA storage where each symbol encodes a distribution over nucleotides and analyzes two central problems: the coverage depth (how many reads are needed) to decode composite strands, and the design of composite alphabets that maximize decoding success under maximum likelihood decoding. It introduces the composite DNA channel, formalizes observed distributions, and derives exact and asymptotic expressions for the coverage depth E(ℓ, ω), including a closed form for ω=2 and a general tail-sum representation for arbitrary ω. The authors then characterize optimal composite alphabets, showing that including the base alphabet and a uniform mixture often yields optimal or near-optimal decoding performance, with explicit results for the case m=q+1 and a detailed binary alphabet treatment. Overall, the paper provides fundamental bounds, exact results for key special cases, and practical guidance for alphabet design in composite DNA storage, highlighting the trade-offs between average and worst-case decoding performance under ML decoding.

Abstract

This paper studies two problems that are motivated by the novel recent approach of composite DNA that takes advantage of the DNA synthesis property which generates a huge number of copies for every synthesized strand. Under this paradigm, every composite symbols does not store a single nucleotide but a mixture of the four DNA nucleotides. The first problem studies the expected number of strand reads in order to decode a composite strand or a group of composite strands. In the second problem, our goal is study how to carefully choose a fixed number of mixtures of the DNA nucleotides such that the decoding probability by the maximum likelihood decoder is maximized.
Paper Structure (7 sections, 43 theorems, 335 equations, 9 figures)

This paper contains 7 sections, 43 theorems, 335 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The Composite DNA Channel
  3. The Coverage Depth Problem for Composite DNA
  4. Optimal Composite Alphabets
  5. Analysis of the Maximum Likelihood Decoder
  6. Basic Optimal Alphabets
  7. Results for the Binary Alphabet

Key Result

Theorem 1

For any $\ell\in\mathbb{N}$ it holds that

Figures (9)

  • Figure 1: All $2$-composite pairs over $(\mathsf{A},\mathsf{C},\mathsf{G},\mathsf{T})$.
  • Figure 2: An illustration of the sequence $\boldsymbol{s}$.
  • Figure 3: An illustration for a $5$-transmission of $\boldsymbol{s}$.
  • Figure 4: The changing observed set of the $5$-transmission.
  • Figure 5: The value of $E(\ell,2)$ (blue) and $\log_2(\ell)+2\frac{1}{3}$ (red) as a function of $\ell$. The difference in green on logarithmic scale.
  • ...and 4 more figures

Theorems & Definitions (87)

  • Example 1
  • Example 2
  • Example 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • ...and 77 more