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

For a binary deletion channel operating on fixed-length inputs, the relevant figure of merit is the finite-blocklength capacity $C_N(d)=\max_{p(x^N)}\mathrm{I}(X^N;Y)$, not only the infinite-blocklength limit $C(d)$. We show that an optimal input may be chosen constant on complement and permutation-