On near-MDS codes and caps

Abstract

Several classes of near-MDS codes of $\mathrm{PG}(3,q)$ are described. They are obtained either by considering the intersection of an elliptic quadric ovoid and a Suzuki-Tits ovoid of a symplectic polar space $\mathcal{W}(3,q)$ or starting from the $q+1$ points of a twisted cubic of $\mathrm{PG}(3,q)$. As a by-product two classes of complete caps of $\mathrm{PG}(4,q)$ of size $2q^2-q\pm \sqrt{2q}+2$ are exhibited.