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A subset $\mathcal{S}$ of a conic $\mathcal{C}$ in the projective plane$\mathrm{PG}(2,q)$ is called almost complete (AC-subset for short) if it can beextended 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. New upper bounds on the smallest size $t(q)$ of an AC-subset areobtained, in particular, \begin{align*} &t(q)<\sqrt{q(3\ln q+\ln\ln q+\ln3)}+\sqrt{\frac{q}{3\ln q}}+4\thicksim\sqrt{3q\ln q};&t(q)<1.835\sqrt{q\lnq}.\end{align*} The new bounds are used to increase regions of pairs $(N,q)$for which it is proved that every normal rational curve in $\mathrm{PG}(N,q)$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$[q+2,N+1,q-N+2]_q$ MDS code.

On Almost Complete Subsets of a Conic in $\mathrm{PG}(2,q)$, Completeness of Normal Rational Curves and Extendability of Reed-Solomon Codes