The DNA Coverage Depth Problem: Duality, Weight Distributions, and Applications

TL;DR

This paper introduces combinatorial tools to analyze the DNA coverage depth problem for various linear codes, providing formulas that relate coverage depth to code weight distributions and extending understanding beyond MDS codes.

Contribution

It develops new combinatorial methods based on duality and weight enumerators to compute coverage depth for multiple linear codes, including non-MDS codes, over small fields.

Findings

Derived closed-form formulas for coverage depth of specific codes

Established a general expression linking coverage depth to weight distributions of code extensions

Enhanced understanding of code performance in DNA data storage applications

Abstract

The coverage depth problem in DNA data storage is about computing the expected number of reads needed to recover all encoded strands. Given a generator matrix of a linear code, this quantity equals the expected number of randomly drawn columns required to obtain full rank. While MDS codes are optimal when they exist, i.e., over large fields, practical scenarios may rely on structured code families defined over small fields. In this work, we develop combinatorial tools to solve the DNA coverage depth problem for various linear codes, based on duality arguments and the notion of extended weight enumerator. Using these methods, we derive closed formulas for the simplex, Hamming, ternary Golay, extended ternary Golay, and first-order Reed-Muller codes. The centerpiece of this paper is a general expression for the coverage depth of a linear code in terms of the weight distributions of its higher-field extensions.