The Capacity of the Weighted Read Channel

Nanopore sequencing is emerging as a powerful tool for storing digital information in DNA molecules. Its advantages over traditional sequencing methods such as support for long reads, scalability, and relatively low cost, making it an attractive and growing area of research. A fundamental aspect of this technology is the nature of its output: instead of reading individual bases in isolation, the sequencer observes overlapping segments of the DNA sequence due to inter-symbol interference (ISI). Similar readout mechanisms also arise in emerging memory technologies such as domain wall (racetrack) memory, where transverse-read techniques can sense the number of stored bits within a region without shifting any domains. Motivated by this analogy, we study theoretical models that capture the fundamental structure of such overlapping read processes. In this work, we study a simplified model of the nanopore sequencing process, known as the (ℓ, δ)-read channel. This channel operates by applying a sliding window of length ℓ across a DNA sequence and shifting it by δ positions after each read. The output of each window is a numerical value, which, in our model, corresponds to the sum (or Hamming weight) of the bases within that window. The collection of these outputs forms what we refer to as theread vector. Due to overlapping windows, multiple input sequences can map to the same read vector, leading to ambiguity. The main object of interest in this work is thecapacityof the (ℓ, δ)-read channel, defined as the number of distinct output sequences that can be generate by this channel. Previous studies have determined the capacity for specific combinations of ℓ and δ, such as when ℓ ≤ δ, when ℓ is a multiple of δ, and for certain small values of ℓ and δ. In this study, we extend the current knowledge by addressing more general cases. Our first main result is a closed-form expression for the capacity when δ2( 1/2 (ℓ + 1 + √(ℓ + 1)2− 4(ℓ − δ)(ℓ − δ + 1) )). This result is derived by analyzing a deterministic finite-state representation of the read channel and applying tools from graph theory, particularly the Perron–Frobenius Theorem. Next, we consider the case where ℓ ≥ 2δ and δ does not divide ℓ. For this setting, we construct a constrained code that yields an upper bound on the capacity, improving upon known bounds and further refining our understanding of this more complex regime. Finally, we extend the model to the two-dimensional setting. Motivated by the physical layout of some sequencing platforms and potential future applications, we propose and analyze a two-dimensional version of the weighted read channel and we present initial results on its capacity.