Distance-regular graph with intersection array {143,108,27;1,12,117} does not exist

Abstract

There is a formally self-dual distance-regular graph $\Gamma$ with classical parameters
$d=3$, $b=\alpha+1=q$, $\beta=q^2+q-1$ and intersection array
$\{(q^2+q-1)(q^2+q+1),(q^2+q)q^2,q^3;1,(q^2+q),q^2(q^2+q+1)\}$. For the graph $\Gamma$ we have the strongly regular graphs
$\Gamma_2$ and $\Gamma_3$
($\Gamma_3$ is pseuqo-geometric for $pG_{q-1}(q^2+q-1,(q^2+q+1)(q-1))$).

It is proved that a distance-regular graph with intersection array $\{143,\\108,27;1,12,117\}$ ($q=3$) does not exist.