Optimal $Z$ -Complementary Code Set From Generalized Reed-Muller Codes

Z-complementary code set (ZCCS), an extension of perfect CCs, refers to a set of 2-D matrices having zero correlation zone properties. ZCCS can be used in various multi-channel systems to support, for example, quasi-synchronous interference-free multicarrier code-division multiple access communication and optimal channel estimation in multiple-input multiple-output systems. Traditional constructions of ZCCS heavily rely on a series of sequence operations which may not be feasible for rapid hardware generation particularly for long ZCCSs. In this paper, we propose a direct construction of ZCCS using the second-order Reed-Muller codes with efficient graphical representation. Our proposed construction, valid for any number of isolated vertices present in the graph, is capable of generating optimal ZCCS meeting the set size upper bound.