Optimal and Suboptimal Quadriphase Sequences Derived from Maximal Length Sequences over Z _{{\bf 4}}

Abstract

The paper presents families of quadriphase sequences derived from maximal length sequences over {\bf Z}_4 having good correlation properties. These are: (i) families of quadriphase sequences of period (2^r-1) from maximal length sequences over {\bf Z}_4; each family consisting of (2^r+1) sequences, (ii) families of quadriphase sequences of period 2(2^r-1) from interleaved maximal length sequences over {\bf Z}_4; each family consisting of (2^{r-1}+1) sequences. Such sequences are of interest in quadriphase modulated code division multiple access communication systems, where it is desirable to have large sets of sequences that possess low value of \theta _\max }, the maximum magnitude of the periodic crosscorrelation and out of phase auto-correlation values. The sequences over {\bf Z}_4 are viewed as trace functions of appropriately chosen unit elements of Galois extension rings of {\bf Z}_4. Quadriphase sequences are then obtained from {\bf Z}_4 sequences, by a quadriphase mapping, \Pi, from {\bf Z}_4 to 4^th roots of unity, given by, \Pi (x) = \omega ^x; where x \in {\bf Z}_4 and \omega = \sqrt-1. Periodic correlation properties (correlation values and their distribution) of the quadriphase sequences are obtained by using an Abelian association scheme on the elements of the corresponding Galois extension ring of {\bf Z}_4. The majority of the families of sequences derived are optimal with respect to the Welch lower bound on \theta _\max; the rest being suboptimal with \theta _\max bounded by \sqrt2L, where L is the period\newpage of the sequences. However nearly half of the sequences in these families are balanced.