Determine the maximum size of partial ovoids of the elliptic quadric

Determine the exact value of the maximum cardinality \(\mu^-(5,q)\) of a partial ovoid of the elliptic quadric \(Q^-(5,q)\) for the remaining field sizes, in order to obtain corresponding exact or improved bounds for symmetric \(2\)-codes in \(\mathrm{Sym}_3(q)\).

Background

The paper establishes a correspondence between partial ovoids of the elliptic quadric $Q^-(5,q)$ and symmetric $2$-codes in $\mathrm{Sym}_3(q)$. Specifically, a partial ovoid of size $m$ yields a symmetric $2$-code in $\mathrm{Sym}_3(q)$ of size $q^2(m-1)$, and the largest code obtainable as a union of complete cosets of the constant-rank subspace $H_\delta$ has size $q^2(\mu^-(5,q)-1)$.

The exact value of $\mu^-(5,q)$ is known for some small parameters, including $q=3$ and $q=5$, but remains undetermined for the other field sizes listed in the paper. Consequently, resolving the partial-ovoid extremal problem would directly improve the available lower bounds for non-additive symmetric rank-metric codes.