Let $X$ be the Fermat hypersurface of dimension $2m$ and of degree $q+1$ defined over an algebraically closed field of characteristic $p>0$, where $q$ is a power of $p$, and let $NL^m (X)$ be the free abelian group of numerical equivalence classes of linear subspaces of dimension $m$ contained in $X$. By the intersection form, we regard $NL^m (X)$ as a lattice. Investigating the configuration of these linear subspaces, we show that the rank of $NL^m (X)$ is equal to the $2m$th Betti number of $X$, that the intersection form multiplied by $(-1)^m$ is positive definite on the primitive part of $NL^m (X)$, and that the discriminant of $NL^m (X)$ is a power of~$p$. Let ${\mathcal L}^m (X)$ be the primitive part of $NL^m (X)$ equipped with the intersection form multiplied by $(-1)^m$. In the case $p=q=2$, the lattice ${\mathcal L}^m (X)$ is described in terms of certain codes associated with the unitary geometry over ${\mathbb F}_2$. Since ${\mathcal L}^1 (X)$ is isomorphic to the root lattice of type $E_6$, the series of lattices ${\mathcal L}^m (X)$ can be considered as a generalization of $E_6$. The lattice ${\mathcal L}^2 (X)$ is isomorphic to the laminated lattice of rank $22$. This isomorphism explains Conway's identification $\cdot 222\cong {\rm PSU}(6,2)$ geometrically. The lattice ${\mathcal L}^3 (X)$ is of discriminant $2^{16}\cdot 3$, minimal norm $8$, and kissing number $109421928$. 2000 Mathematics Subject Classification: 14C25, 11H31, 51D25.