We introduce a "W*"-metric space, which is a particular approach to non-commutative metric spaces where a "quantum metric" is defined on a von Neumann algebra. We generalize the notion of a quantum code and quantum error correction to the setting of finite dimensional "W*"-metric spaces, which includes codes and error correction for classical finite metric spaces. We also introduce a class of "W*"-metric spaces that come from representations of semi-simple Lie algebras [g] called [g]-"metric" spaces, and present an outline for code constructions. In turn, we produce specific code constructions for [su] (2,[C])-metric spaces that depend upon proving Tverberg's theorem for points on a moment curve constructed from arithmetic sequences. We introduce a "quantum distance distribution," and we prove an analogue of the MacWilliam's identities for [su] (2)-metric spaces. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]