The Capacity of the Weighted Read Channel

One of the primary sequencing methods gaining prominence in DNA storage is nanopore sequencing, attributed to various factors. In this work, we consider a simplified model of the sequencer, characterized as a channel. This channel takes a sequence and processes it using a sliding window of length $\ell$, shifting the window by $\delta$ characters each time. The output of this channel, which we refer to as the read vector, is a vector containing the sums of the entries in each of the windows. The capacity of the channel is defined as the maximal information rate of the channel. Previous works have already revealed capacity values for certain parameters $\ell$ and $\delta$. In this work, we show that when $\delta < \ell < 2\delta$, the capacity value is given by $\frac{1}{\delta}\log_2 \frac{1}{2}(\ell+1+ \sqrt{(\ell+1)^2 - 4(\ell - \delta)(\ell-\delta +1)})$. Additionally, we construct an upper bound when $2\delta < \ell$. Finally, we extend the model to the two-dimensional case and present several results on its capacity.