On the Smallest Size of an Almost Complete Subset of a Conic in PG(2, q) and Extendability of Reed–Solomon Codes

Abstract

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,

$$t(q) < \sqrt {q(3lnq + lnlnq + ln3)} + \sqrt {\frac{q}{{3\ln q}}} + 4 \sim \sqrt {3q\ln q} ,t(q) < 1.835\sqrt {q\ln q.} $$

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, qN + 2]q maximum distance separable code.

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Correspondence to D. Bartoli.

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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

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