Minimal gauge invariant couplings at order $\alpha'^3$: NS-NS fields

Removing the field redefinitions, the Bianchi identities and the total derivative freedoms from the general form of gauge invariant NS-NS couplings at order $\alpha'^3$, we have found that the minimum number of independent couplings is 872. We find that there are schemes in which there is no term with structures $R,\,R_{\mu\nu},\,\nabla_\mu H^{\mu\alpha\beta}$, $\nabla_\mu\nabla^\mu\Phi$. In these schemes, there are sub-schemes in which, except one term, the couplings can have no term with more than two derivatives. In the sub-scheme that we have chosen, the 872 couplings appear in 55 different structures. We fix some of the parameters in type II supersting theory by its corresponding four-point functions. The coupling which has term with more than two derivatives is constraint to be zero by the four-point functions.