12. Homophonic Sequence Substitution

Homophonic sequence substitution is the name given in this paper to the technique,
which consists of substituting one-to-one a given finite or semi-infinite sequence
of symbols by another finite or semi-infinite sequence over the same alphabet,
but having a higher entropy rate. The output sequence of a given discrete stationary and
ergodic source is encoded a binary lossless source code C. A concatenation of codewords of C
is then conveniently parsed and reencoded with a binary lossless source code.
By iterating the latter step a number of times, it is proved that the entropy rate of the
binary sequence at the output of the last encoder approaches the value one asymptotically,
therefore performing optimum homophonic sequence substitution.
The remaining redundancy, after k consecutive encodings, is 1 minus H sub k of S
bits per binary digits, where H sub k is the entropy rate of the binary sequence
resulting after the kth encoding. A Markov source model is presented to describe the
binary encoded sequences and to compute their entropy rate.