Linear codes associated with the Desarguesian ovoids in $Q^{+}(7,q)$

Abstract

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 $q$.