Constructions of Polyphase Golay Complementary Arrays

TL;DR

This paper extends Golay complementary matrices to multi-dimensional arrays, introduces new construction methods using algebraic identities, and explores size feasibility for different array types, advancing applications in omnidirectional precoding.

Contribution

It generalizes GCM to GCA, proposes new construction techniques, and establishes size feasibility conditions for quaternary and binary GCA pairs and quads.

Findings

Quaternary GCA pairs are feasible if the product of sizes is a quaternary Golay number.

Binary GCM quads likely have arbitrary feasible sizes, supported by partial verification.

All positive integers up to 1000 are feasible sizes for quaternary GCM quads in one dimension.

Abstract

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.