On the Divisibility of Discrete Sources with an Additive Single-Letter Distortion Measure

Hierarchical $\varepsilon$-nets for discrete metric spaces as well as hierarchical multilevel codes for discrete memoryless sources are considered. A divisibility problem is formulated. The divisibility means that in each level of the hierarchy, the total amount of the information necessary to pass from an achieved distortion level to a smaller one asymptotically coincides with the corresponding increment of the $\varepsilon$-entropy of the space or the rate-distortion function of the source. Conditions under which a ternary equiprobable source with the balanced distortion measure is divisible are found.