On the Capacity of Noisy Frequency-based Channels

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.