Generalized GMW quadriphase sequences satisfying the Elch bound with equality (Q1973899)

This paper gives new families of quadriphase sequences obtained from large linear complexity sequences over \(\mathbb{Z}_4\) by making use of generalized permutation monomials over Galois rings. The construction of these sequences can be seen as a generalization of the binary GMW sequence construction, and hence they are referred to as GGMW sequences over \(\mathbb{Z}_4\). The GGMW families satisfy the Welch bound on inner products with equality, and it is shown that the root mean square of all the crosscorrelations and out-of-phase autocorrelations \((\theta_{\text{rms}})\) is approximately equal to the quantity \(\sqrt L; L\) being the period of the sequences. However, \(\theta_{\text{max}}\), the maximum magnitude of periodic crosscorrelation and out-of-phase autocorrelation, deviates from the optimal value of \(\sqrt L\). Computer results suggest that the number of crosscorrelation values which deviate from the optimal value of \(\sqrt L\) is small. The weight structure of these sequences is the same as those of \(m\)-sequences over \(\mathbb{Z}_4\). The linear complexity (LC) of the sequences is computed using a generalized Blahuts theorem on the LC of sequences over \(\mathbb{Z}_4\).