Optimality of Dimension Expanding Shannon-Kotel´nikov Mappings

Shannon-Kotel\´nikov mappings are joint source-channel coding (JSCC) systems, realized as direct source-channel mappings. In this paper we show that dimension expanding Shannon-Kotel\´nikov mappings can get as close as we want to the information theoretical bounds (OPTA) by letting the source and channel dimension go towards infinity. We use our own generalization of Kotel\´nikov\´s theory on 1:N bandwidth expanding modulation systems, and the "sphere hardening" effect (a consequence of the law of large numbers) for this purpose. At the end of the paper we give a rough estimate on how close one can get to OPTA for a given dimensionality.