  The Desarguesian ovoids in the orthogonal polar space $Q^+(7,q)$ with $q$
even have first been introduced by Kantor by examining the $8$-dimensional
absolutely irreducible modular representations of $\text{PGL}(2,q^3)$. We
investigate this module for all prime power values of $q$. The shortest
$\text{PGL}(2,q^3)$-orbit $O$ gives the Desarguesian ovoid in $Q^+(7,q)$ for
even $q$ and it is known to give a complete partial ovoid of the symplectic
polar space $W(7,q)$ for odd~$q$. We determine the hyperplane sections of $O$.
As a corollary, we obtain the parameters $[q^3+1,8,q^3-q^2-q]_q$ and the weight
distribution of the associated $\mathbb{F}_q$-linear code $C_O$ and the
parameters $[q^3+1,q^3-7,5]_q$ of the dual code $C_O^\perp$ for $q \ge 4$. We
also show that both codes $C_O$ and $C_O^\perp$ are length-optimal for all
prime power values of $q$.
