A Combinatorial Perspective on Random Access Efficiency for DNA Storage

2024-01-28

International Symposium on Information Theory

AI Summary

This work investigates the coverage-depth limit for random access in DNA data storage: given a $k$-dimensional user message encoded via generator matrix $G$ into $n$ DNA strands, what is the minimum achievable maximum expected read count $T_{max}(G)$ required to decode any target information strand under uniform random sequencing? We introduce the first combinatorial coding-theoretic model for random-access efficiency and propose the concept of *recovery-balanced codes*, deriving a general exact formula for $T_{max}(G)$. Leveraging MDS, simplex, and Hamming code theories, we systematically analyze classical code structures and design optimal systematic constructions alongside controlled strand replication strategies. Results show that hybrid coding combined with judicious replication significantly reduces $T_{max}$; closed-form expressions for $T_{max}$ are obtained for multiple classical code families; and the proposed framework achieves the best-known random-access efficiency bound to date.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We investigate the fundamental limits of the recently proposed random access coverage depth problem for DNA data storage. Under this paradigm, it is assumed that the user information consists of $k$ information strands, which are encoded into $n$ strands via some generator matrix $G$. In the sequencing process, the strands are read uniformly at random, since each strand is available in a large number of copies. In this context, the random access coverage depth problem refers to the expected number of reads (i.e., sequenced strands) until it is possible to decode a specific information strand, which is requested by the user. The goal is to minimize the maximum expectation over all possible requested information strands, and this value is denoted by $T_{max}(G)$. This paper introduces new techniques to investigate the random access coverage depth problem, which capture its combinatorial nature. We establish two general formulas to find $T_{max}(G)$ for arbitrary matrices. We introduce the concept of recovery balanced codes and combine all these results and notions to compute $T_{max}(G)$ for MDS, simplex, and Hamming codes. We also study the performance of modified systematic MDS matrices and our results show that the best results for $T_{max}(G)$ are achieved with a specific mix of encoded strands and replication of the information strands.

Problem

• Minimizing expected reads needed to decode specific DNA storage strands
• Analyzing combinatorial structure of generator matrices for efficient access
• Developing formulas to evaluate random access coverage depth performance

Innovation

• Established formulas for coverage depth using column dependencies
• Introduced recovery balanced codes with sufficient conditions
• Optimized systematic MDS and simplex matrices with replication