Paper Info
Title
On sets without tangents and exterior sets of a conic
Abstract
A set without tangents in PG(2,q) is a set of points S such that no line meets S in exactly one point. An exterior set of a conic C is a set of points E such that all secant lines of E are external lines of C. In this paper, we first recall some known examples of sets without tangents and describe them in terms of determined directions of an affine pointset. We show that the smallest sets without tangents in PG(2,5) are (up to projective equivalence) of two different types. We generalise the non-trivial type by giving an explicit construction of a set without tangents in PG(2,q), q=p^h, p2 prime, of size q(q-1)/2-r(q+1)/2, for all 0@?r@?(q-5)/2. After that, a different description of the same set in PG(2,5), using exterior sets of a conic, is given and we investigate in which ways a set of exterior points on an external line L of a conic in PG(2,q) can be extended with an extra point Q to a larger exterior set of C. It turns out that if q=3 mod 4, Q has to lie on L, whereas if q=1 mod 4, there is a unique point Q not on L.
Year
DOI
Venue
2011
10.1016/j.disc.2011.07.010
Discrete Mathematics
Keywords
DocType
Volume
exterior set of a conic,ldpc code,set without tangents,stopping sets
Journal
311
Issue
ISSN
Citations 
20
G. Electronic Journal of Combinatorics. 17 (1) (2010), R174
0
PageRank 
References 
Authors
0.34
7
2
Name
Order
Citations
PageRank
van de voorde1357.85
geertrui210.70