On graphs and codes associated to the sporadic simple groups $\mathrm{HS}$ and $\mathrm{M}_{22}$

We provide an alternative construction of the strongly regular graphs associated with the two sporadic simple groups $\mathrm{M}_{22}$ and $\mathrm{HS}$. Further, we give some new constructions of other known strongly regular graphs by taking the orbits of a certain subgroup of $\mathrm{M}_{22}$ on the planes of the Hermitian variety $\mathcal{H}(5,4)$. These geometric constructions can be used to produce cap codes with large parameters and automorphism groups containing $\mathrm{M}_{22}$ as a subgroup.