On the Generality of $1+\mathbf{i}$ as a Non-Norm Element

Full-rate space-time block codes with nonvanishing determinants have been extensively designed with cyclic division algebras. For these designs, smaller pairwise error probabilities of maximum likelihood detections require larger normalized diversity products, which can be obtained by choosing integer non-norm elements with smaller absolute values. All known methods have constructed $1+\bi$ and $2+\bi$ to be integer non-norm elements with the smallest absolute values over QAM for the number of transmit antennas $n$: $\{n:5\leq n\leq 40,8\nmid n\}$ and $\{n:5\leq n\leq 40,8\mid n\}$, respectively. Via explicit constructions, this paper proves that $1+\bi$ is an integer non-norm element with the smallest absolute value over QAM for every $n\geq 5$.