On the smallest size of an almost complete subset of a conic in $\mathrm{PG}(2,q)$ and extendability of Reed–Solomon codes

In the projective plane $\mathrm{PG}(2,q)$, a subset $\mathcal S$ of a conic $\mathcal C$ is said to be almost complete if it can be extended to a larger arc in $\mathrm{PG}(2,q)$ only by the points of $\mathcal C\setminus\mathcal S$ and by the nucleus of $\mathcal C$ when $q$ is even. We obtain new upper bounds on the smallest size $t(q)$ of an almost complete subset of a conic, in particular,
$$ \begin{aligned} & t(q)<\sqrt{q(3\ln q+\ln\ln q+\ln3)}+\sqrt{\frac q{3\ln q}}+4\sim\sqrt{3q\ln q},\\ & t(q)<1{,}835\sqrt{q\ln q}. \end{aligned} $$
The new bounds are used to extend the set of pairs $(N,q)$ for which it is proved that every normal rational curve in the projective space $\mathrm{PG}(N,q)$ is a complete $(q+1)$-arc, or equivalently, that no $[q+1,N+1,q-N+1]_q$ generalized doubly-extended Reed–Solomon code can be extended to a $[q+2,N+1,q-N+2]_q$ maximum distance separable code.