Optimal Codes for the Coverage Depth Problem and the Performance of Random Codes

2026-10-05

AI Summary

This study addresses the problem of minimizing the expected number of random sampling reads required to cover the information space in DNA storage, with particular focus on optimal code construction when MDS codes are unavailable. By leveraging combinatorial mathematics, independent set analysis of dual codes, and the log-concavity of matroid basis generating polynomials, the authors derive a general formula for the expected coverage depth. This work establishes the first theoretical framework for optimal codes under non-MDS parameters, proving the optimality of simplex and Hamming codes. Furthermore, it precisely quantifies the expected coverage depth of random linear codes, revealing that their gap from the optimal benchmark remains bounded at constant rates.

Abstract

DNA storage systems retrieve information by randomly sampling synthesized DNA strands, making the number of reads required for successful recovery a fundamental performance measure. This motivates the coverage depth problem: for given code parameters, determine a linear code that minimizes the expected number of randomly sampled columns required to span the entire information space. While MDS codes are known to be optimal whenever they exist, identifying optimal codes in parameter regimes where MDS codes do not exist remains largely open. In this work, we derive a general formula for the expectation in the coverage depth setting, and apply this result to establish the optimality of the $q$-ary simplex code for the parameters that allow its existence. Moreover, we show the optimality of the $q$-ary Hamming code by relating coverage depth to independent sets in the dual code and exploiting log-concavity properties of matroid basis-generating polynomials. We further analyze random linear codes, derive an exact expression for their expected coverage depth, and show that, in the constant-rate regime, their additive gap from the MDS benchmark remains bounded independently of the code length.

Problem

• DNA storage
• coverage depth problem
• linear codes
• MDS codes
• optimal codes

Innovation

• Coverage Depth Problem
• DNA Storage
• Simplex Code
• Hamming Code
• Random Linear Codes