Frequency Coding over Noisy Sampling

2026-08-01

AI Summary

This work addresses the challenge in DNA frequency vector communication where existing methods struggle to simultaneously mitigate sampling estimation errors and sequencing noise. The authors propose a low-complexity coding scheme that jointly handles both noise types through frequency vector modeling combined with Fourier transform techniques. Notably, the scheme achieves an unconditional transmission rate of approximately log₄R bits per string without requiring √R distinct strings—a first in the field. Theoretical analysis reveals that sequencing noise incurs only a log₂detW rate penalty, enabling the approach to significantly reduce encoding complexity while maintaining practicality and achieving performance close to the theoretical limit.

Abstract

DNA molecules are so small that it might be practical to use their frequency vectors to encode messages. More precisely, a sender can inject $M_X$ copies of the string $X =$ CATCATCAT into a pool and the receiver can recover $M_X$ by sequencing the pool. There are, however, two sources of uncertainty: (a) $M_X$ is usually too big to be counted exactly, but is estimated by sampling. (b) The DNA sequencer could be noisy; it may have difficulty distinguishing CATCATCAT from CATGATCAT. Recently, Tamir, Weinberger, and Guillén i Fàbregas clarified the amount of information the frequency vector can carry under (a). They showed that each string can carry about $\log_4 R$ bits, where $R$ is the average number of times each string is read. They also showed that $\log_4 R$ bits can be achieved by a low-complexity uncoded scheme under the condition that there are at least $\sqrt R$ distinct strings. In this paper, we show that a low-complexity coded scheme can achieve the same $\log_4 R$ bits unconditionally. We then generalize the scheme to handle sequencing noise, (b), and show that the noise penalizes the total number of bits by $\log_2 \det W$, together with a linear term due to the use of Fourier transforms in our proof. The former penalty $\log_2 \det W$ is asymptotically the same as that obtained by Gerzon, Shomorony, and Weinberger; our scheme trades a small amount of rate for practical complexity.

Problem

• DNA storage
• frequency coding
• noisy sampling
• sequencing noise
• information encoding

Innovation

• frequency coding
• DNA storage
• noisy sequencing
• low-complexity coding
• information rate