On the Capacity of Noisy Frequency-based Channels
Yuval Gerzon, Ilan Shomorony, Nir Weinberger
TL;DR
This work analyzes the capacity of noisy frequency-based channels motivated by DNA storage in the short-molecule regime, where information is encoded in molecule frequencies rather than order. It extends prior noiseless results by introducing a Markov-noise model for identification errors and deriving a vector Poisson-channel-based achievability bound, together with a converse bound via data processing. A key contribution is a transparent capacity penalty term $\Delta = \frac{1}{2n}\log\det(WW^T)$ (or $\log|\det(W)|$ in the square case) that captures the degradation due to identification noise, with explicit forms for DNA-relevant noise models (erasure and symmetric substitution) and for the DNA channel $W=w^{\otimes L}$. The results quantify how identification noise reduces the scaling of reliably stored bits and identify a regime for $\beta$ (e.g., $\beta > \frac{2}{3\log|\mathcal A|}$) under which the analysis holds, informing practical design for DNA-based archival storage and guiding extensions to broader noise models.
Abstract
We investigate the capacity of noisy frequency-based channels, motivated by DNA data storage in the short-molecule regime, where information is encoded in the frequency of items types rather than their order. The channel output is a histogram formed by random sampling of items, followed by noisy item identification. While the capacity of the noiseless frequency-based channel has been previously addressed, the effect of identification noise has not been fully characterized. We present a converse bound on the channel capacity that follows from stochastic degradation and the data processing inequality. We then establish an achievable bound, which is based on a Poissonization of the multinomial sampling process, and an analysis of the resulting vector Poisson channel with inter-symbol interference. This analysis refines concentration inequalities for the information density used in Feinstein bound, and explicitly characterizes an additive loss in the mutual information due to identification noise. We apply our results to a DNA storage channel in the short-molecule regime, and quantify the resulting loss in the scaling of the total number of reliably stored bits.