Distance-regular graph with intersection array $\{140,108,18;1,18,105\}$ does not exist

Distance-regular graph $\Gamma$ of diameter $3$ having the second eigenvalue $\theta_1=a_3$ is called Shilla graph. In this case $a=a_3$ devides $k$ and we set $b=b(\Gamma)=k/a$. Jurishich and Vidali found intersection arrays of $Q$-polynomial Shilla graphs with $b_2=c_2$: $\{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\}$. But many arrays in this series are not feasible. Belousov I. N. and Makhnev A. A. found a new infinite series feasible arrays of $Q$-polynomial Shilla graphs with $b_2=c_2$ ($t=2r^2-1$): $\{2r(2r^2-1)(2r+1),(2r-1)(2r(2r^2-1)+2r^2),r(2r^2+r-1);1,r(2r^2+r-1),(2r^2-1)(4r^2-1)\}$. If $r=2$ then we have intersection array $\{140,108,18;1,18,105\}$. In the paper it is proved that graph with this intersection array does not exist.