Lower bounds on the maximum cross-correlations of 2-D quasi-complementary array sets

For one-dimensional (1-D) sequences, many lower bounds on the maximum cross-correlations have been demonstrated. For example, bounds proposed by Welch, Levenstein, Liu et al., and others are the lower bounds on the maximum cross-correlations of aperiodic 1-D sequence sets or quasi-complementary sequence sets (QCSSs). However, in recent times, two-dimensional (2-D) arrays have emerged with promising applications in wireless communication, such as ultra wide-band (UWB), 2-D synchronization, massive multiple-input multiple-output (MIMO), 2-D multi-carrier code division multiple access (2D-MC-CDMA), etc. Although the construction of a 2-D quasi-complementary array set (QCAS) exists in literature, the lower bound on the maximum cross-correlation \(\delta _{max}\) of such a 2-D QCAS has not been reported previously. In this paper, we propose, for the first time lower bounds on the maximum cross-correlations of 2-D QCASs for both periodic and aperiodic cases. The existing lower bounds on the maximum cross-correlations of 1-D QCSSs and 1-D sequence sets can be deduced from the proposed lower bounds on the maximum cross-correlations of 2-D QCASs and 2-D array sets for certain cases.