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

Daniele Bartoli, Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco

Published 2016-09-19Version 1

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 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. New upper bounds on the smallest size $t(q)$ of an AC-subset are obtained, in particular \begin{align*} & t(q)<\sqrt{q(3\ln q+\ln\ln q +\ln3)}+\sqrt{\frac{q}{3\ln q}}+4\thicksim \sqrt{3}\sqrt{q\ln q};\\ & t(q)<1.835\sqrt{q\ln q}. \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 a $[q+1,N+1,q-N+1]_q$ generalized doubly-extended Reed-Solomon code cannot be extended to a $[q+2,N+2,q-N+1]_q$ code.