Quotients of Orders in Cyclic Algebras and Space-Time Codes

Abstract

Let $F$ be a number field with ring of integers $\Oc_F$ and $\Dc$ a division $F$-algebra with a maximal cyclic subfield $K$. We study rings occurring as quotients of a natural $\Oc_F$-order $\Lambda$ in $\Dc$ by two-sided ideals. We reduce the problem to studying the ideal structure of $\Lambda/\qf^s\Lambda$, where $\qf$ is a prime ideal in $\Oc_F$, $s\geq 1$. We study the case where $\qf$ remains unramified in $K$, both when $s=1$ and $s>1$. This work is motivated by its applications to space-time coded modulation.