ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 3, pp. 217–223.
Pleiades Publishing, Inc., 2011.
Original Russian Text
R. Daskalov, E. Metodieva, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 3, pp. 3–9.
New (n, r)-Arcs in PG(2, 17), PG(2, 19),
and PG(2, 23)
R. Daskalov and E. Metodieva
Department of Mathematics, Technical University of Gabrovo, Bulgaria
Received May 20, 2010
Abstract—An (n, r)-arc is a set of n points of a projective plane such that some r but no r +1
of them are collinear. The maximum size of an (n, r)-arc in PG(2,q) is denoted by m
In this paper a new (95, 7)-arc, (183, 12)-arc, and (205, 13)-arc in PG(2, 17) are constructed,
as well as a (243, 14)-arc and (264, 15)-arc in PG(2, 19). Likewise, good large (n, r)-arcs in
PG(2, 23) are constructed and a table with bounds on m
(2, 23) is presented. In this way many
new 3-dimensional Griesmer codes are obtained. The results are obtained by nonexhaustive
local computer search.
Let GF (q) denote the Galois ﬁeld of q elements, and let V (3,q) be the vector space of row
vectors of length 3 with entries in GF (q). Let PG(2,q) be the corresponding projective plane.
) of PG(2,q) are 1-dimensional subspaces of V (3,q). Subspaces of dimension two
are called lines. The number of points and the number of lines in PG(2,q)isq
+ q +1. Thereare
q + 1 points on every line and q + 1 lines through every point.
Deﬁnition 1. An (n, r)-arc is a set of n points of a projective plane such that some r but no
r + 1 of them are collinear.
The maximum size of an (n, r)-arc in PG(2,q) is denoted by m
Deﬁnition 2. An (l, t)-blocking set S in PG(2,q)isasetofl points such that every line of
PG(2,q) intersects S in at least t points and there is a line intersecting S in exactly t points.
Note that an (n, r)-arc is the complement of a (q
+q +1−n, q+1−r)-blocking set in a projective
plane, and vice versa.
Deﬁnition 3. Let M be a set of points in any plane. An i-secant is a line meeting M in
exactly i points. Deﬁne τ
as the number of i-secants to a set M.
In terms of τ
the deﬁnitions of an (n, r)-arc and (l, t)-blocking set become as follows.
Deﬁnition 4. An (n, r)-arc is a set of n points of a projective plane for which τ
> 0, and τ
Deﬁnition 5. An (l, t)-blocking set is a set of l points of a projective plane for which τ
> 0, and τ
The following two theorems are proved in [1, 2], respectively.
Supported in part by the Bulgarian Ministry of Education and Science under contract in TU-Gabrovo.