ISSN 0032-9460, Problems of Information Transmission, 2014, Vol. 50, No. 4, pp. 320–339.
Pleiades Publishing, Inc., 2014.
Original Russian Text
D. Bartoli, A.A. Davydov, G. Faina, A.A. Kreshchuk, S. Marcugini, F. Pambianco, 2014, published in Problemy Peredachi
Informatsii, 2014, Vol. 50, No. 4, pp. 22–42.
Upper Bounds on the Smallest Size of a Complete Arc
in PG(2,q) under a Certain Probabilistic Conjecture
, and F. Pambianco
Department of Mathematics and Computer Sciences,
Universit`a degli Studi di Perugia, Italy
e-mail: email@example.com, firstname.lastname@example.org, email@example.com,
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
e-mail: firstname.lastname@example.org, email@example.com
Received April 19, 2014; in ﬁnal form, August 25, 2014
Abstract—In the projective plane PG(2,q), we consider an iterative construction of complete
arcs which adds a new point in each step. It is proved that uncovered points are uniformly
distributed over the plane. For more than half of steps of the iterative process, we prove an
estimate for the number of newly covered points in every step. A natural (and well-founded)
conjecture is made that the estimate holds for the other steps too. As a result, we obtain upper
bounds on the smallest size t
(2,q) of a complete arc in PG(2,q), in particular,
3lnq +lnlnq +ln3+
(2,q) < 1.87
q ln q.
Nonstandard types of upper bounds on t
(2,q) are considered, one of them being new. The
eﬀectiveness of the new bounds is illustrated by comparing them with the smallest known sizes
of complete arcs obtained in recent works of the authors and in the present paper via computer
search in a wide region of q. We note a connection of the considered problems with the so-called
birthday problem (or birthday paradox).
Let PG(2,q) be the projective plane over the Galois ﬁeld of q elements. An n-arc is a set
of n points no three of which are collinear. An n-arc is said to be complete if it is contained in no
(n + 1)-arc in PG(2,q). For an introduction to projective geometries over ﬁnite ﬁelds, see [1–3].
Relationships between the theory of n-arcs, coding theory, and mathematical statistics are pre-
sented in [4, 5] (see also ). In particular, a complete arc in PG(2,q), the points of which are
treated as 3-dimensional q-ary columns, deﬁnes a parity-check matrix of a q-ary linear code with
codimension 3, Hamming distance 4, and covering radius 2. Arcs can be interpreted as linear
Supported in part by the Ministry for Education, University and Research of Italy (MIUR), project
“Geometrie di Galois e strutture di incidenza,” and Italian National Group for Algebraic and Geometric
Structures and their Applications (G.N.S.A.G.A.).
Supported by the European Community under a Marie-Curie Intra-European Fellowship, FACE project