A normal rational curve of the <inline-formula> <tex-math notation="LaTeX">$(k-1)$ </tex-math></inline-formula>-dimensional projective space over <inline-formula> <tex-math notation="LaTeX">${\mathbb F}_{q}$ </tex-math></inline-formula> is an arc of size <inline-formula> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula>, since any <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> points of the curve span the whole space. In this paper, we will prove that if <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is odd, then a subset of size <inline-formula> <tex-math notation="LaTeX">$3k-6$ </tex-math></inline-formula> of a normal rational curve cannot be extended to an arc of size <inline-formula> <tex-math notation="LaTeX">$q+2$ </tex-math></inline-formula>. In fact, we prove something slightly stronger. Suppose that <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is odd and <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula> is a <inline-formula> <tex-math notation="LaTeX">$(2k-3)$ </tex-math></inline-formula>-subset of an arc <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> of size <inline-formula> <tex-math notation="LaTeX">$3k-6$ </tex-math></inline-formula>. If <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> projects to a subset of a conic from every <inline-formula> <tex-math notation="LaTeX">$(k-3)$ </tex-math></inline-formula>-subset of <inline-formula> <tex-math notation="LaTeX">$E$ </tex-math></inline-formula>, then <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> cannot be extended to an arc of size <inline-formula> <tex-math notation="LaTeX">$q+2$ </tex-math></inline-formula>. Stated in terms of error-correcting codes we prove that a <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-dimensional linear maximum distance separable code of length <inline-formula> <tex-math notation="LaTeX">$3k-6$ </tex-math></inline-formula> over a field <inline-formula> <tex-math notation="LaTeX">${\mathbb F}_{q}$ </tex-math></inline-formula> of odd characteristic, which can be extended to a Reed–Solomon code of length <inline-formula> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula>, cannot be extended to a linear maximum distance separable code of length <inline-formula> <tex-math notation="LaTeX">$q+2$ </tex-math></inline-formula>.
[1]
Simeon Ball.
Finite Geometry and Combinatorial Applications: References
,
2015
.
[2]
Ron M. Roth,et al.
On MDS extensions of generalized Reed-Solomon codes
,
1986,
IEEE Trans. Inf. Theory.
[3]
O. Antoine,et al.
Theory of Error-correcting Codes
,
2022
.
[4]
Simeon Ball,et al.
On sets of vectors of a finite vector space in which every subset of basis size is a basis II
,
2012,
Designs, Codes and Cryptography.
[5]
Leo Storme.
Completeness of Normal Rational Curves
,
1992
.
[6]
J. Hirschfeld,et al.
The packing problem in statistics, coding theory and finite projective spaces : update 2001
,
2001
.
[7]
Ameera Chowdhury.
Inclusion Matrices and the MDS Conjecture
,
2016,
Electron. J. Comb..
[8]
J. Thas,et al.
General Galois geometries
,
1992
.
[9]
W. Marsden.
I and J
,
2012
.
[10]
Alexander Vardy.
What's New and Exciting in Algebraic and Combinatorial Coding Theory?
,
2006,
ISIT.
[11]
Simeon Ball.
Extending small arcs to large arcs
,
2016,
1603.05795.