Orbit Reduction and Learned Run Distributions for Finite-Blocklength Binary Deletion Channels

Abstract

In DNA data storage, racetrack memories, and packet networks, data are written as many short strands of fixed length. The receiver knows where each strand begins and ends, and deletions occur only inside a strand. The relevant limit for such systems is the block capacity $C_N(d)$, the largest mutual information between a length-$N$ input and the output of a binary deletion channel with deletion probability $d$. Because the boundaries are known, $C_N(d)/N$ is never smaller than the classical capacity $C(d)$. Computing $C_N(d)$ is hard because the input takes $2^N$ values. For $0\le d<1$, we show that the optimal input is unique, gives positive probability to every string, and is unchanged by complementing or reversing the strings. We then introduce the optimized run distribution (ORD), which assigns one probability to each number of runs; it is optimal for $N\le3$ and is solved with a numerical optimality certificate up to $N=16$. For longer strands, an exact recursion for the output distribution gives unbiased rate estimates, and an empirical Bernstein inequality turns them into confidence intervals. Within the statistical precision, a one-parameter Markov input performs as well as the best inputs found. For strands of 100-200 bits, known boundaries increase the rate by up to 0.026 bits per symbol beyond the best certified upper bound on the capacity of the unsegmented channel. Via Fano's inequality, the block capacity also bounds the rate of any code that uses a single strand; in the tabulated case, this bound is tighter than the best known finite-length bound for strands longer than about 90 bits. Conversely, block rates yield lower bounds on $C(d)$; at $d=0.1$, the bound is within $1.1\times10^{-4}$ bits/use of the best certified lower bound. Finally, InfoNCE estimates, even with the optimal critic, empirically lose most of their ability to rank inputs as the strand length grows.