  A polynomial indicator function of designs is first introduced by Fontana
{\it et al}. (2000) for two-level cases. They give the structure of the
indicator functions, especially the relation to the orthogonality of designs.
These results are generalized by Aoki (2019) for general multi-level cases. As
an application of these results, we can enumerate all orthogonal fractional
factorial designs with given size and orthogonality using computational
algebraic software. For example, Aoki (2019) gives classifications of
orthogonal fractions of 2x2x2x2x3 designs with strength 3, which is derived by
simple eliminations of variables. However, the computational feasibility of
this naive approach depends on the size of the problems. In fact, it is
reported that the computation of orthogonal fractions of 2x2x2x2x3 designs with
strength 2 fails to carry out in Aoki (2019). In this paper, using the theory
of primary decomposition, we enumerate and classify orthogonal fractions of
2x2x2x2x3 designs with strength 2. We show there are 35200 orthogonal half
fractions of 2x2x2x2x3 designs with strength 2, classified into 63 equivalent
classes.
