  The formal multiple zeta space we consider with a computer is an
$\mathbb{F}_2$-vector space generated by $2^{k-2}$ formal symbols for a given
weight $k$, where the symbols satisfy binary extended double shuffle relations.
Up to weight $k=22$, we compute the dimensions of the formal multiple zeta
spaces, and verify the dimension conjecture on original extended double shuffle
relations of real multiple zeta values. Our computations adopt Gaussian forward
elimination and give information for spaces filtered by depth. We can observe
that the dimensions of the depth-graded formal multiple zeta spaces have a
Pascal triangle pattern expected by the Hoffman mult-indices.
