Cohomology rings of oriented Grassmann manifolds ${\tilde{G}}_{2^t,4}$

We give a description of the mod 2 cohomology algebra of the oriented Grassmann manifold ${\tilde{G}}_{2^t,4}$ as the quotient of a polynomial algebra by a certain ideal. In the process we find a Gröbner basis for that ideal, which we then use to exhibit an additive basis for $H^{\ast }({\tilde{G}}_{2^t,4};{\mathbb{Z}}_2)$.