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Coded Information Retrieval for Block-Structured DNA-Based Data Storage

Daniella Bar-Lev

Abstract

We study the problem of coded information retrieval for block-structured data, motivated by DNA-based storage systems where a database is partitioned into multiple files that must each be recoverable as an atomic unit. We initiate and formalize the block-structured retrieval problem, wherein $k$ information symbols are partitioned into two files $F_1$ and $F_2$ of sizes $s_1$ and $s_2 = k - s_1$. The objective is to characterize the set of achievable expected retrieval time pairs $\bigl(E_1(G), E_2(G)\bigr)$ over all $[n,k]$ linear codes with generator matrix $G$. We derive a family of linear lower bounds via mutual exclusivity of recovery sets, and develop a nonlinear geometric bound via column projection. For codes with no mixed columns, this yields the hyperbolic constraint $s_1/E_1 + s_2/E_2 \le 1$, which we conjecture to hold universally whenever $\max\{s_1,s_2\} \ge 2$. We analyze explicit codes, such as the identity code, file-dedicated MDS codes, and the systematic global MDS code, and compute their exact expected retrieval times. For file-dedicated codes we prove MDS optimality within the family and verify the hyperbolic constraint. For global MDS codes, we establish dominance by the proportional local MDS allocation via a combinatorial subset-counting argument, providing a significantly simpler proof compared to recent literature and formally extending the result to the asymmetric case. Finally, we characterize the limiting achievability region as $n \to \infty$: the hyperbolic boundary is asymptotically achieved by file-dedicated MDS codes, and is conjectured to be the exact boundary of the limiting achievability region.

Coded Information Retrieval for Block-Structured DNA-Based Data Storage

Abstract

We study the problem of coded information retrieval for block-structured data, motivated by DNA-based storage systems where a database is partitioned into multiple files that must each be recoverable as an atomic unit. We initiate and formalize the block-structured retrieval problem, wherein information symbols are partitioned into two files and of sizes and . The objective is to characterize the set of achievable expected retrieval time pairs over all linear codes with generator matrix . We derive a family of linear lower bounds via mutual exclusivity of recovery sets, and develop a nonlinear geometric bound via column projection. For codes with no mixed columns, this yields the hyperbolic constraint , which we conjecture to hold universally whenever . We analyze explicit codes, such as the identity code, file-dedicated MDS codes, and the systematic global MDS code, and compute their exact expected retrieval times. For file-dedicated codes we prove MDS optimality within the family and verify the hyperbolic constraint. For global MDS codes, we establish dominance by the proportional local MDS allocation via a combinatorial subset-counting argument, providing a significantly simpler proof compared to recent literature and formally extending the result to the asymmetric case. Finally, we characterize the limiting achievability region as : the hyperbolic boundary is asymptotically achieved by file-dedicated MDS codes, and is conjectured to be the exact boundary of the limiting achievability region.
Paper Structure (21 sections, 32 theorems, 109 equations, 4 figures)

This paper contains 21 sections, 32 theorems, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Encoding Model and File Structure
  4. Subspace Geometries and Column Counting
  5. Motivation and Problem Formulation
  6. Linear Lower Bounds
  7. Basic lower bounds
  8. Combinatorial Cuts from Mutual Exclusivity
  9. A Nonlinear Bound via Column Geometry
  10. Projected retrieval times and the main theorem
  11. The universal hyperbolic conjecture
  12. Constructions and Finite-$n$ Achievability
  13. The identity code
  14. File-dedicated codes
  15. Systematic $[n,k]$ MDS codes
  16. ...and 6 more sections

Key Result

Lemma 1

For any generator matrix $G \in \mathbb{F}_q^{k \times n}$ and target index set $I \subseteq [k]$, the expected number of uniform draws with replacement to recover all coupons in $I$ is given by:

Figures (4)

  • Figure 1: Combinatorial cut bounds for $k=8$, $n=20$, across four partition configurations $(s_1,s_2)$. The solid line is the tightest cut $s^*=k-1$; dashed lines show weaker cuts. Black dots are local MDS operating points.
  • Figure 2: Same as \ref{['fig:cuts_n20']} but for $n=50$. The cuts tighten noticeably and the local MDS points approach the cut boundaries.
  • Figure 3: Nonlinear bounds for $k=8$, $n=20$, across four partition configurations $(s_1,s_2)$. Solid lines are proved bounds (Corollaries \ref{['cor:smax_bound']} and \ref{['cor:cs_bound']}); the dashed line is the conjectured hyperbolic boundary (Conjecture \ref{['conj:hyperbola']}). Black dots are local MDS operating points. The gray region is provably non-achievable.
  • Figure 4: Same as \ref{['fig:nonlinear_n20']} but for $n=50$. The local MDS points approach the conjectured hyperbolic boundary as $n$ grows, consistent with Theorem \ref{['thm:asymptotic_ub']}.

Theorems & Definitions (79)

  • Definition 1: Recovery Set gruica2024combinatorial
  • Definition 2: Subset Count gruica2024combinatorial
  • Lemma 1: gruica2024combinatorial
  • Definition 3: File Retrieval Time
  • Definition 4: Column Counting Function
  • Lemma 2
  • proof
  • Lemma 3: abraham2024covering
  • Example 1
  • Example 2
  • ...and 69 more