On instantaneous codes for zero-error coding of two correlated sources

We study the problem of finding zero-error instantaneous codes for the Slepian-Wolf configuration. By a zero-error instantaneous code we mean two encoder maps f/sub 1/:/spl chi//spl rarr/{0,1}*, f/sub 2/:Y/spl rarr/{0,1}* and a decoding algorithm which, for any pair of encoder outputs f/sub 1/(x/sub 1/)f/sub 1/(x/sub 2/)f/sub 1/(x/sub 3/)... and f/sub 2/(y/sub 1/)f/sub 2/(y/sub 2/)f/sub 2/(y/sub 3/)..., can correctly determine x/sub 1/ and y/sub 1/ by reading only the first length (f/sub 1/(x/sub 1/)) bits of the X encoder output and the first length (f/sub 2/(y/sub 1/)) bits of the Y encoder output. For |/spl chi/|=2, we find a necessary and sufficient condition for the existence of a zero-error instantaneous code with a given set of codeword lengths for Y. Using this condition, we derive an upper bound to the minimum expected codeword length for Y and construct a simple example showing that the coding scheme proposed by Kh, Jabri and Al-Issa (1997) is not optimal in general, and more surprisingly, the optimal code may violate the "Morse condition" that the more probable of two symbols never has the longer codeword. For |/spl chi/|=3, we find a necessary but not sufficient condition for the existence of a zero-error instantaneous code with a given set of codeword lengths for Y. Moreover, for |/spl chi/|/spl ges/3, the existence of such a code is shown to be related to a rectangle packing problem.