  In this work, binary block-to-block distribution matching is considered. m
independent and uniformly distributed bits are mapped to n output bits
resembling a target product distribution. A rate R is called achieved by a
sequence of encoder-decoder pairs, if for m,n to infinity, (1) m/n approaches
R, (2) the informational divergence per bit of the output distribution and the
target distribution goes to zero, and (3) the probability of erroneous decoding
goes to zero. It is shown that the maximum achievable rate is equal to the
entropy of the target distribution. A practical encoder-decoder pair is
constructed that provably achieves the maximum rate in the limit. Numerical
results illustrate that the suggested system operates close to the limits with
reasonable complexity. The key idea is to internally use a fixed-to-variable
length matcher and to compensate underflow by random mapping and to cast an
error when overflow occurs.
