Optimal and suboptimal quadriphase sequences derived from maximal length sequences over \(\mathbb{Z}_4\)
Quadriphase sequences are studied with respect to their correlation properties. A quadriphase sequence is a sequence consisting of 4th roots of unity. A set of such sequences is called optimal if the maximum magnitude \(\theta_{\max}\) of the periodic crosscorrelation and out-of-phase auto-correlation meets the Welch bound, and suboptimal if \(\theta_{\max}\) is bounded by \(\sqrt{2L}\), where \(L\) is the length of the sequence. Quadriphase sequences are related to \(\mathbb{Z}_4\)-sequences by using the map \(x\to \omega^x\) where \(\omega\) is a 4th root of unity. It turns out that taking certain (interleaved) maximum length sequences over \(\mathbb{Z}_4\) one can get (suboptimal) quadriphase sequences. These maximum length sequences over \(\mathbb{Z}_4\) are constructed using the trace function from the Galois extension ring GR\((4,r)\) to \(\mathbb{Z}_4\). The correlation properties are obtained by examining the eigenvalues of an Abelian association scheme on GR\((4,r)\) and by using the linearity of the trace function.
scientific article; zbMATH DE number 1195712