On perfect colorings of the halved 24-cube

We consider 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 the 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 $c=1,2,4,5,7$. Tabl. 2, bibl. 4.