  We consider conditional exact tests of factor effects in designed experiments
for discrete response variables. Similarly to the analysis of contingency
tables, Markov chain Monte Carlo methods can be used for performing exact
tests, especially when large-sample approximations of the null distributions
are poor and the enumeration of the conditional sample space is infeasible. To
construct a connected Markov chain over the appropriate sample space, a common
approach is to compute a Markov basis. Theoretically, a Markov basis can be
characterized as a generator of a well-specified toric ideal in a polynomial
ring and is computed by computational algebraic softwares. However, the
computation of a Markov basis sometimes becomes infeasible even for problems of
moderate sizes. In this paper, we obtain the closed form expression of minimal
Markov bases for the main effect models of $2^{p-1}$ fractional factorial
designs of resolution $p$.
