  Pseudorandom number generators are required to generate pseudorandom numbers
which have good statistical properties as well as unpredictability in
cryptography. An m-sequence is a linear feedback shift register sequence with
maximal period over a finite field. M-sequences have good statistical
properties, however we must nonlinearize m-sequences for cryptographic
purposes. A geometric sequence is a sequence given by applying a nonlinear
feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a
geometric sequence whose nonlinear feedforward function is given by the
Legendre symbol, and showed the period, periodic autocorrelation and linear
complexity of the sequence. Furthermore, Nogami et al. proposed a
generalization of the sequence, and showed the period and periodic
autocorrelation. In this paper, we first investigate linear complexity of the
geometric sequences. In the case that the Chan--Games formula which describes
linear complexity of geometric sequences does not hold, we show the new formula
by considering the sequence of complement numbers, Hasse derivative and
cyclotomic classes. Under some conditions, we can ensure that the geometric
sequences have a large linear complexity from the results on linear complexity
of Sidel'nikov sequences. The geometric sequences have a long period and large
linear complexity under some conditions, however they do not have the balance
property. In order to construct sequences that have the balance property, we
propose interleaved sequences of the geometric sequence and its complement.
Furthermore, we show the periodic autocorrelation and linear complexity of the
proposed sequences. The proposed sequences have the balance property, and have
a large linear complexity if the geometric sequences have a large one.
