An Improved Number-Theoretic Method for Constructing Approximately Universal Codes

Cyclic division algebras have been extensively utilized to design approximately universal codes. Their error probabilities can be intrinsically improved by a small integer non-norm element. This paper proves the insufficiency of existing encoding methods for obtaining the non-norm element 1+i over QAM and the numbers of transmit antennas: {n:8|n}. An improved method is then presented to overcome this difficulty by number theory. As a by-product, we show that 1+i is one of the smallest integer non-norm elements over QAM and n≥5.