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Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over thefinite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$.The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ withrespect to the orbits of points and planes under the action of the stabilizergroup of the twisted cubic is described. This information is used to viewgeneralized doubly-extended Reed-Solomon codes of codimension four asasymptotically optimal multiple covering codes.

On planes through points off the twisted cubic in $\mathrm{PG}(3,q)$ and multiple covering codes