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

TL;DR

This paper establishes new upper bounds on the size of almost complete subsets of conics in projective planes, which in turn improves understanding of the completeness of normal rational curves and the extendability of Reed-Solomon codes.

Contribution

It introduces novel bounds on the minimal size of AC-subsets in projective planes, impacting the theory of normal rational curves and Reed-Solomon code extendability.

Findings

New upper bounds on AC-subset sizes, approximately proportional to 3535353535353535353535353535353535",

The bounds improve the known regions where normal rational curves are complete arcs.

The results have implications for the extendability of certain Reed-Solomon codes.

Abstract

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{3q\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 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.