Strong optimality conjecture for the Desarguesian ovoid code

Let $C_O$ be the code associated with the Desarguesian ovoid in $Q^+(7,q)$, with $q\geq 3$. A code is strongly optimal when it is both $P$-optimal and $S$-optimal. Strong optimality conjecture. The code $C_O$ is strongly optimal for all $q\geq 3$. The paper proves strong optimality for $q\in\{3,4\}$ and notes that for $q=5$ it would follow from the nonexistence of a linear $[127,8,45]_5$ code, whose existence is open; the conjecture remains open in general.