Divergence Scaling of Fixed-Length, Binary-Output, One-to-One Distribution Matching

Distribution matching is the process of invertibly mapping a uniformly distributed input sequence onto sequences that approximate the output of a desired discrete memoryless source. The special case of a binary output alphabet and one-to-one mapping is studied. A fixed-length distribution matcher is proposed that is optimal in the sense of minimizing the unnormalized informational divergence between its output distribution and a binary memoryless target distribution. Upper and lower bounds on the unnormalized divergence are computed that increase logarithmically in the output block length $n$. It follows that a recently proposed constant composition distribution matcher performs within a constant gap of the minimal achievable informational divergence.