TY  - JOUR
Y1  - 2015///
VL  - 48
AV  - restricted
JF  - EUROPEAN JOURNAL OF COMBINATORICS
N2  - A finite k-net of order n is an incidence structure consisting of k >= 3 pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the k classes. Deleting a line class from a k-net, with k >= 4, gives a derived (k 1)-net of the same order. Finite k-nets embedded in a projective plane PG(2, K) coordinatized by a field K of characteristic 0 only exist for k = 3, 4, see Korchmaros et al. (2014). In this paper, we investigate 3-nets embedded in PG(2, K) whose line classes are in perspective position with an axis r, that is, every point on the line r incident with a line of the net is incident with exactly one line from each class. The problem of determining all such 3-nets remains open whereas we obtain a complete classification for those coordinatizable by a group. As a corollary, the (unique) 4-net of order 3 embedded in PG(2, K) turns out to be the only 4-net embedded in PG(2, K) with a derived 3-net which can be coordinatized by a group. Our results hold true in positive characteristic under the hypothesis that the order of the k-net considered is smaller than the characteristic of K. (C) 2015 Elsevier Ltd. All rights reserved.
UR  - https://doi.org/10.1016/j.ejc.2015.02.019
EP  - 185
SN  - 0195-6698
A1  -  Bogya Norbert
A1  -  Korchmaros G
A1  -  Nagy Gábor Péter
N1  - Cited By :3            
            Export Date: 1 June 2023            
            CODEN: EJOCD
ID  - publicatio29129
TI  - Classification of k-nets
SP  - 177
ER  -