Division Algebras and Wireless Communication

T he aim of this note is to bring to the attentionof awidemathematicalaudience the recent application of division algebras to wireless communication. The application occurs in the context of communication involving multiple transmit and receive antennas, a context known in engineering as MIMO—short for multiple input, multiple output. While the use of multiple receive antennas goes back to the time of Marconi, the basic theoretical framework for communication using multiple transmit antennas was only published about ten years ago. The progress in the field has been quite rapid, however, and MIMO communication is widely credited with being one of the key emerging areas in telecommunication. Our focus here will be on one aspect of this subject: the formatting of transmit information for optimum reliability. Recall that a division algebra is an (associative) algebra with a multiplicative identity in which every nonzero element is invertible. The center of a division algebra is the set of elements in the algebra that commute with every other element in the algebra; the center is itself just a commutative field, and the division algebra is naturally a vector space over its center. We consider only division