Exact Error Exponents of Concatenated Codes for DNA Storage

TL;DR

This paper derives the exact error exponents for concatenated DNA storage codes, demonstrating improvements over previous bounds and analyzing various scaling regimes with implications for code design.

Contribution

It provides the first exact error exponent analysis for concatenated DNA storage codes and introduces a coded-index strategy that achieves optimal error exponents.

Findings

Derived exact error exponents in key regimes

Showed coded-index strategies attain highest error exponents

Analyzed error behavior in super-linear and low-rate regimes

Abstract

In this paper, we consider a concatenated coding based class of DNA storage codes in which the selected molecules are constrained to be taken from an ``inner'' codebook associated with the sequencing channel. This codebook is used in a ``black-box'' manner, and is only assumed to operate at an achievable rate in the sense of attaining asymptotically vanishing maximal (inner) error probability. We first derive the exact error exponent in a widely-studied regime of constant rate and a linear number of sequencing reads, and show strict improvements over an existing achievable error exponent. Moreover, our achievability analysis is based on a coded-index strategy, implying that such strategies attain the highest error exponents within the broader class of codes that we consider. We then extend our results to other scaling regimes, including a super-linear number of reads, as well as several certain low-rate regimes. We find that the latter comes with notable intricacies, such as the suboptimality of codewords with all distinct molecules, and certain dependencies of the error exponents on the model for sequencing errors.