How Much Information Can Be Stored Using Very Short DNA Molecules?

From an information-theoretic point of view, the commonly adopted DNA storage channel, the shuffling-sampling channel, has two distinct operational regimes. If the stored molecules are relatively long, so that the number of molecule types is larger than the number of molecules that store the message, the channel capacity is positive. Alternatively, if the molecules are relatively short and so that the number of molecule types is smaller than the number of molecules that store the message, then the resulting storage system is characterized by capacity zero. In this short molecule regime, Shomorony and Heckel (2022) put forward a conjecture on the scaling of the number of information bits that can be reliably stored in the noiseless case. This conjecture was only partially proved by Gerzon, Weinberger, and Shomorony (2025). In the first part of this talk, I will describe a random-coding scheme which completes the proof of the conjecture. Since this random-coding scheme is computationally heavy, the second part of this talk is devoted to an alternative coding scheme which operates at a significantly lower computational complexity but achieves the optimal scaling, except for a specific range of very short molecules. In the final part of this talk, I will briefly discuss the noisy case and describe a concatenated coding scheme which combines linear block codes together with zero-undetected-error decoding.