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On subsets of the normal rational curve

Simeon Ball and Jan De Beule (UGent)
(2017) IEEE TRANSACTIONS ON INFORMATION THEORY. 63(6). p.3658-3662
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Abstract
A normal rational curve of the (k - 1)- dimensional projective space over F-q is an arc of size q+1, since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k-6 of a normal rational curve cannot be extended to an arc of size q+2. In fact, we prove something slightly stronger. Suppose that q is odd and E is a (2k - 3)-subset of an arc G of size 3k - 6. If G projects to a subset of a conic from every (k - 3)- subset of E, then G cannot be extended to an arc of size q + 2. Stated in terms of error-correcting codes we prove that a k-dimensional linear maximum distance separable code of length 3k - 6 over a field Fq of odd characteristic, which can be extended to a Reed-Solomon code of length q + 1, cannot be extended to a linear maximum distance separable code of length q + 2.
Keywords
MDS code, normal rational curve, Reed-Solomon code

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MLA
Ball, Simeon, and Jan De Beule. “On Subsets of the Normal Rational Curve.” IEEE TRANSACTIONS ON INFORMATION THEORY, vol. 63, no. 6, 2017, pp. 3658–62, doi:10.1109/TIT.2017.2671344.
APA
Ball, S., & De Beule, J. (2017). On subsets of the normal rational curve. IEEE TRANSACTIONS ON INFORMATION THEORY, 63(6), 3658–3662. https://doi.org/10.1109/TIT.2017.2671344
Chicago author-date
Ball, Simeon, and Jan De Beule. 2017. “On Subsets of the Normal Rational Curve.” IEEE TRANSACTIONS ON INFORMATION THEORY 63 (6): 3658–62. https://doi.org/10.1109/TIT.2017.2671344.
Chicago author-date (all authors)
Ball, Simeon, and Jan De Beule. 2017. “On Subsets of the Normal Rational Curve.” IEEE TRANSACTIONS ON INFORMATION THEORY 63 (6): 3658–3662. doi:10.1109/TIT.2017.2671344.
Vancouver
1.
Ball S, De Beule J. On subsets of the normal rational curve. IEEE TRANSACTIONS ON INFORMATION THEORY. 2017;63(6):3658–62.
IEEE
[1]
S. Ball and J. De Beule, “On subsets of the normal rational curve,” IEEE TRANSACTIONS ON INFORMATION THEORY, vol. 63, no. 6, pp. 3658–3662, 2017.
@article{8566612,
  abstract     = {{A normal rational curve of the (k - 1)- dimensional projective space over F-q is an arc of size q+1, since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k-6 of a normal rational curve cannot be extended to an arc of size q+2. In fact, we prove something slightly stronger. Suppose that q is odd and E is a (2k - 3)-subset of an arc G of size 3k - 6. If G projects to a subset of a conic from every (k - 3)- subset of E, then G cannot be extended to an arc of size q + 2. Stated in terms of error-correcting codes we prove that a k-dimensional linear maximum distance separable code of length 3k - 6 over a field Fq of odd characteristic, which can be extended to a Reed-Solomon code of length q + 1, cannot be extended to a linear maximum distance separable code of length q + 2.}},
  author       = {{Ball, Simeon and De Beule, Jan}},
  issn         = {{0018-9448}},
  journal      = {{IEEE TRANSACTIONS ON INFORMATION THEORY}},
  keywords     = {{MDS code,normal rational curve,Reed-Solomon code}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{3658--3662}},
  title        = {{On subsets of the normal rational curve}},
  url          = {{http://doi.org/10.1109/TIT.2017.2671344}},
  volume       = {{63}},
  year         = {{2017}},
}
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