Abstract—In the projective plane PG(2, q), a subset S of a conic C is said to be almost complete if it can be extended to a larger arc in PG(2, q) only by the points of C \ S and by the nucleus of C when q is even. We obtain new upper bounds on the smallest size t(q) of an almost complete subset of a conic, in particular,
The new bounds are used to extend the set of pairs (N, q) for which it is proved that every normal rational curve in the projective space 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 maximum distance separable code.
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Hirschfeld, J.W.P., Projective Geometries over Finite Fields, Oxford: Clarendon; New York: Oxford Univ. Press, 1998, 2nd ed.
Hirschfeld, J.W.P. and Storme, L., The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001, Finite Geometries (Proc. 4th Isle of Thorns Conf., July 16–21, 2000), Blokhuis, A., Hirschfeld, J.W.P., Jungnickel, D., and Thas, J.A., Eds., Dev. Math., vol. 3, Dordrecht: Kluwer, 2001, pp. 201–246.
Hirschfeld, J.W.P. and Thas, J.A., Open Problems in Finite Projective Spaces, Finite Fields Appl., 2015, vol. 32, no. 1, pp. 44–81.
Ball, S., Finite Geometry and Combinatorial Applications, Cambridge, UK: Cambridge Univ. Press, 2015.
Ball, S. and De Beule, J., On Subsets of the Normal Rational Curve, arXiv:1603.06714 [math.CO], 2016.
Chowdhury, A., Inclusion Matrices and the MDS Conjecture, arXiv:1511.03623v3 [math.CO], 2015.
Hirschfeld, J.W.P., Korchmáros, G., and Torres, F., Algebraic Curves over a Finite Field, Princeton: Princeton Univ. Press, 2008.
Klein, A. and Storme, L., Applications of Finite Geometry in Coding Theory and Cryptography, Information Security, Coding Theory and Related Combinatorics, Crnković, D. and Tonchev, V., Eds., NATO Sci. Peace Secur. Ser. D: Inf. Commun. Secur., vol. 29. Amsterdam: IOS Press, 2011, pp. 38–58.
Landjev, I. and Storme, L., Galois Geometry and Coding Theory, Current Research Topics in Galois Geometry, De Beule, J. and Storme, L., Eds., New York: Nova Science Pub., 2011, ch. 8, pp. 185–212.
MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.
Roth, R.M., Introduction to Coding Theory, Cambridge: Cambridge Univ. Press, 2007.
Storme, L. and Thas, J.A., Complete k-Arcs in PG(n, q), q Even, Discrete Math., 1992, vol. 106/107, pp. 455–469.
Storme, L. and Thas, J.A., k-Arcs and Dual k-Arcs, Discrete Math., 1994, vol. 125, no. 1–3, pp. 357–370.
Thas, J.A., M.D.S. Codes and Arcs in Projective Spaces: A Survey, Matematiche (Catania), 1992, vol. 47, no. 2, pp. 315–328.
Segre, B., Curve razionali normali e k-archi negli spazi finiti, Ann. Mat. Pura Appl., 1955, vol. 39, no. 1, pp. 357–379.
Ball, S., On Sets of Vectors of a Finite Vector Space in Which Every Subset of Basis Size is a Basis, J. Eur. Math. Soc., 2012, vol. 14, no. 3, pp. 733–748.
Ball, S. and De Beule, J., On Sets of Vectors of a Finite Vector Space in Which Every Subset of Basis Size is a Basis. II, Des. Codes Cryptogr., 2012, vol. 65, no. 1, pp. 5–14.
Korchmáros, G., Storme, L., and Szőnyi, T., Space-Filling Subsets of a Normal Rational Curve, J. Statist. Plann. Inference, 1997, vol. 58, no. 1, pp. 93–110.
Storme, L., Completeness of Normal Rational Curves, J. Algebraic Combin., 1992, vol. 1, no. 2, pp. 197–202.
Storme, L. and Thas, J.A., Generalized Reed–Solomon Codes and Normal Rational Curves: An Improvement of Results by Seroussi and Roth, Advances in Finite Geometries and Designs (Proc. 3rd Isle of Thorns Conf., Chelwood Gate, UK, 1990), Hirschfeld, J.W.P., Hughes, D.R., and Thas, J.A., Eds., Oxford: Oxford Univ. Press, 1991, pp. 369–389.
Kovács, S.J., Small Saturated Sets in Finite Projective Planes, Rend. Mat. Appl., 1992, vol. 12, no. 1, pp. 157–164.
Bosma, W., Cannon, J., and Playoust, C., The Magma Algebra System, I: The User Language, J. Symbolic Comput., 1997, vol. 24, no. 3–4, pp. 235–265.
Bartoli, D., Davydov, A.A., Faina, G., Kreshchuk, A.A., Marcugini, S., and Pambianco, F., Upper Bounds on the Smallest Size of a Complete Arc in a Finite Desarguesian Projective Plane Based on Computer Search, J. Geom., 2016, vol. 107, no. 1, pp. 89–117.
Bartoli, D., Davydov, A.A., Marcugini, S., and Pambianco, F., On Almost Complete Subsets of a Conic in PG(2, q), Completeness of Normal Rational Curves and Extendability of Reed–Solomon Codes, arXiv:1609.05657v3 [math.CO], 2017.
Original Russian Text © D. Bartoli, A.A. Davydov, S. Marcugini, F. Pambianco, 2018, published in Problemy Peredachi Informatsii, 2018, Vol. 54, No. 2, pp. 3–19.
The research has been carried out using computing resources of the Federal Collective Usage Center “Complex for Simulation and Data Processing for Mega-science Facilities” at the National Research Center “Kurchatov Institute,” http://ckp.nrcki.ru/.
Supported in part by the Ministry of Education, Universities and Research of Italy (MIUR), project “Geometrie di Galois e strutture di incidenza”, Italian National Group for Algebraic and Geometric Structures and Their Applications (GNSAGA–INDAM), and University of Perugia, projects “Configurazioni geometriche e superfici altamente simmetriche” and “Codici lineari e strutture geometriche correlate,” Base Research Fund 2015.
The research was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.
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Bartoli, D., Davydov, A.A., Marcugini, S. et al. On the Smallest Size of an Almost Complete Subset of a Conic in PG(2, q) and Extendability of Reed–Solomon Codes. Probl Inf Transm 54, 101–115 (2018). https://doi.org/10.1134/S0032946018020011