  We propose a variation of Construction A of lattices from linear codes
defined using the quotient $\Lambda/\mathfrak p\Lambda$ of some order $\Lambda$
inside a cyclic division $F$-algebra, for $\mathfrak p$ a prime ideal of a
number field $F$. To obtain codes over this quotient, we first give an
isomorphism between $\Lambda/\mathfrak p\Lambda$ and a ring of skew
polynomials. We then discuss definitions and basic properties of skew
polynomial codes, which are needed for Construction A, but also explore further
properties of the dual of such codes. We conclude by providing an application
to space-time coding, which is the original motivation to consider cyclic
division $F$-algebras as a starting point for this variation of Construction A.
