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
{ι_x}
is that for some
0<α<1
Σ/xϵF(2-
l
_x
)≤α for each clique
F
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-log
₂
α bits to the asymptotic limits for each graph in 𝒢, 2) there exist graph classes of interest for which such α can indeed be found, and finally (3) no such α can be found for the class of all graphs.