  The time delay interferometry (TDI) is an algorithm proposed to suppress the
laser frequency noise in space-borne gravitational wave detectors. As a
post-processing technique, it is implemented by constructing a virtual equal
arm interferometer through an appropriate combination of the time-shifted data
streams. Such an approach is tailored to the intrinsic feature of the
space-based gravitational wave detection, namely, the distances between
spacecraft are governed by the orbit dynamics and thus can not be held
constant. Among different implementations, the geometric TDI was introduced as
a method of exhaustion to evaluate the second-generation TDI combinations. The
applications of the algebraic approach based on computational algebraic
geometry, on the other hand, are mostly restricted to the first and the
modified first-generation TDI. Besides, geometric TDI furnishes an intuitive
physical interpretation about the synthesis of the virtual optical paths. In
this paper, the geometric TDI is utilized to investigate the modified
second-generation TDI combinations in conjunction with a ternary search
algorithm. The distinction between the second-generation and modified
second-generation TDI solutions is elaborated regarding the rate of change of
the arm lengths with respect to the opposite cyclic directions. For the
sixteen-link combinations, forty second-generation TDI solutions are recovered,
among which nine of them are identified as the modified second-generation ones.
