Optimal coding with a single standard run length

The optimum single standard run lengths for a binary first-order Markov source are derived and extended to multilevel first-order Markov sources. Maximization of the compression ratio is used as the criterion of optimality. When the output symbols are block coded, the optimal single standard run length
n_i
for each symbol is shown to satisfy an implicit equation of the form
(n_i - 1)(-\ln q_{ii}) = 1 - q_{ii}^ {n_i}
, where
q_{ii}
is a transition probability. An expression for the overall compression ratio is derived for the binary case, and a comparison is made with enumerative source encoding. Compression ratio maxima are found by computer search for the binary independent source when the output symbols are subsequently Huffman coded, and a comparison of this scheme with ordinary run-length and source-extension coding is given.