Kraft Inequality and Zero-Error Source Coding With Decoder Side Information

For certain source coding problems, it is well known that the Kraft inequality provides a simple sufficient condition for the existence of codes with given codeword lengths. Motivated by this fact, a sufficient condition based on the Kraft inequality can also be sought for the problem of zero-error instantaneous coding with decoder side information. More specifically, it can be envisioned that a sufficient condition for the existence of such codes with codeword lengths {l _x} is that for some 0<alphales<1 Sigma_xepsivFscr (2-l x)lesalpha for each clique Fscr in the characteristic graph G of the source-side information pair. In this correspondence, it is shown that (1) if the above is indeed a sufficient condition for a class G of graphs, then it is possible to come as close as 1-log₂ alpha bits to the asymptotic limits for each graph in G, (2) there exist graph classes of interest for which such a can indeed be found, and finally (3) no such n can be found for the class of all graphs.