  A vertex 2-coloring of a graph is said to be perfect with parameters
$(a_{ij})_{i,j=1}^k$ if for every $i,j\in\{1,...,k\}$ every vertex of color $i$
is adjacent with exactly $a_{ij}$ vertices of color $j$. We consider the
perfect 2-colorings of the distance-2 graph of the 24-cube $\{0,1\}^{24}$ with
parameters $((20+c,256-c)(c,276-c))$ (i.e., with eigenvalue 20). We prove that
such colorings exist for all $c$ from 1 to 128 except 1, 2, 4, 5, 7, 10, 13 and
do not exist for $c=1, 2, 4, 5, 7$. Keywords: perfect coloring, equitable
partition, hypercube, halved n-cube
