On sets of plane-type $$(0, mq, 2mq)_2$$ ( 0 , m q , 2 m q ) 2 in $$\mathrm {PG}(3, q)$$ PG ( 3 , q ) with a long
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Journal of Geometry
On sets of plane-type (0, mq, 2mq)2 in PG(3, q) with a long secant Vito Napolitano Abstract. In this paper, we give two chracterizations of sets of plane-type (0, mq, 2mq)2 in PG(3, q). Mathematics Subject Classification. 51E26. Keywords. Projective space, set with few intersection numbers, linear code.
1. Introduction Subsets of points of PG(r, q) such that the range of the sizes of their intersections with all the hyperplanes of the projective space assumes only few values are related with (linear) codes with few weights (cf e.g. [1,2]). This has been one of the motivations of their study and so a wide literature is devoted to such sets. Recently, in [3] the authors study k-sets of points of PG(3, q), q = ph a prime power, intersected by any plane either in 0 or mq or 2mq points. They prove that k = mq 2 , m divides q, and m = q. Finally, they give an example of such a set for q = 2h , i.e. p = 2, and conclude the paper with some questions and observations on the existence of such sets. If q = 2, then m = 1 necessarily and K is a 4-set of plane-type (0, 2, 4)2 , i.e. K is the pointset of two coplanar lines less their common point. So from now on, we will suppose q = 2. In this paper, we show other examples of such sets and we give two characterizations of them under some extra conditions. This research was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM). The author was also supported by the project “VALERE: Vanvitelli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”.
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V. Napolitano
J. Geom.
Let r, h and s be three positive integers and m1 < · · · < ms be s integers, with m1 ≥ 0. Recall, that a set K of points of PG(r, q) is of type (m1 , . . . , ms )h with respect to the family of the h-dimensional subspaces of PG(r, q) if the sizes of the intersections of h-dimensional subsaces of PG(r, q) with K belong to {m1 , . . . , ms } and each integer mi occurs as the size of the intersection of K with a subspace of dimension h of PG(r, q). When h = 1 or 2, we say that K is of line-type (m1 , . . . , ms )1 and plane-type (m1 , . . . , ms )2 , respectively. Also, by a i-line (i-plane) we mean a line (plane) intersecting K in exactly i points. A 0-line ( 0-plane) is called an external line (plane) and an 1-line is a tangent line. It is easy to see that the set of plane-type (0, mq, 2mq)2 described in [3] has a “cone structure”. So, here we give an equivalent description of that set. Example 1. Let q = 2h with h > 1 and m = q/2 = 2h−1 . Let π be a plane of PG(3, 2h ) and let Ω be the pointset of m concurrent lines of π less their common point. Now, let C be the cone projecting the set Ω from a point P0 not on π. It is immediate to see that K = C\{P0 } is an (mq 2 )-set of plane-type (0, mq, 2mq)2 . For m = 1, other examples of such sets of points are known, independently from the characteristic of the field. Example 2. Let K be a (q 2 + q + 1)-set of points of PG(3, q) of plane-type (1, q
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