Polyphase Golay Complementary Arrays: The Possible Sizes and New Constructions

Motivated by the recent application of two-dimensional Golay complementary arrays (2-D GCA) in uniform rectangular array (URA) -based omnidirectional precoding for the next generation of cellular communications, we propose in this paper new constructions of GCA pairs and GCA quads, which allow for more possible sizes. These constructions are facilitated by four identities over a commutative ring, of which two are established results of quaternions and octonions and the others are novel. We prove that the size of a 4-phase GCA pair is possible if the product of the array sizes in all dimensions is a 4-phase Golay number with an additional constraint on the factorization of the product. For 2-phase 2-D GCA quads, all sizes no greater than <inline-formula> <tex-math notation="LaTeX">$78\times 78$ </tex-math></inline-formula> can be covered. For 4-phase GCA quads, all the positive integers within 1000 can be covered for the size in one dimension. Besides producing GCAs that can accommodate to antenna arrays of various sizes, this study also yields new Hadamard matrices of sizes previously unavailable.