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

TL;DR

This paper proves that the element 1+i is the smallest absolute value integer non-norm element over QAM for all transmit antenna counts n ≥ 5, extending previous known cases.

Contribution

It provides an explicit proof that 1+i is the minimal integer non-norm element over QAM for all n ≥ 5, generalizing prior results.

Findings

1+i is the smallest absolute value integer non-norm element for all n ≥ 5.

Extends previous known cases for specific n values.

Provides explicit constructions for these elements.

Abstract

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 $n\geq 5$.