  Golay complementary matrices (GCM) have recently drawn considerable
attentions owing to its potential applications in omnidirectional precoding. In
this paper we generalize the GCM to multi-dimensional Golay complementary
arrays (GCA) and propose new constructions of GCA pairs and GCA quads. These
constructions are facilitated by introducing a set of identities over a
commutative ring. We prove that a quaternary GCA pair is feasible if the
product of the array sizes in all dimensions is a quaternary Golay number with
an additional constraint on the factorization of the product. For the binary
GCM quads, we conjecture that the feasible sizes are arbitrary, and verify for
sizes within 78 $\times$ 78 and other less densely distributed sizes. For the
quaternary GCM quads, all the positive integers within 1000 can be covered for
the size in one dimension.
