2,942 research outputs found
Desarguesian spreads and field reduction for elements of the semilinear group
The goal of this note is to create a sound framework for the interplay
between field reduction for finite projective spaces, the general semilinear
groups acting on the defining vector spaces and the projective semilinear
groups. This approach makes it possible to reprove a result of Dye on the
stabiliser in PGL of a Desarguesian spread in a more elementary way, and extend
it to P{\Gamma}L(n, q). Moreover a result of Drudge [5] relating Singer cycles
with Desarguesian spreads, as well as a result on subspreads (by Sheekey,
Rottey and Van de Voorde [19]) are reproven in a similar elementary way.
Finally, we try to use this approach to shed a light on Condition (A) of
Csajbok and Zanella, introduced in the study of linear sets [4]
On the equivalence of linear sets
Let be a linear set of pseudoregulus type in a line in
, or . We provide examples of
-order canonical subgeometries such
that there is a -space with the property that for , is the projection
of from center and there exists no collineation of
such that and .
Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes
Cryptogr. 56:89-104, 2010) states the existence of a collineation between the
projecting configurations (each of them consisting of a center and a
subgeometry), which give rise by means of projections to two linear sets. It
follows from our examples that this condition is not necessary for the
equivalence of two linear sets as stated there. We characterize the linear sets
for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented.
The final version is to appear in Designs, Codes and Cryptograph
On the number of k-dominating independent sets
We study the existence and the number of -dominating independent sets in
certain graph families. While the case namely the case of maximal
independent sets - which is originated from Erd\H{o}s and Moser - is widely
investigated, much less is known in general. In this paper we settle the
question for trees and prove that the maximum number of -dominating
independent sets in -vertex graphs is between and
if , moreover the maximum number of
-dominating independent sets in -vertex graphs is between
and . Graph constructions containing a large number of
-dominating independent sets are coming from product graphs, complete
bipartite graphs and with finite geometries. The product graph construction is
associated with the number of certain MDS codes.Comment: 13 page
The use of blocking sets in Galois geometries and in related research areas
Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems
A study of (xvt,xvt−1)-minihypers in PG(t,q)
AbstractWe study (xvt,xvt−1)-minihypers in PG(t,q), i.e. minihypers with the same parameters as a weighted sum of x hyperplanes. We characterize these minihypers as a nonnegative rational sum of hyperplanes and we use this characterization to extend and improve the main results of several papers which have appeared on the special case t=2. We establish a new link with coding theory and we use this link to construct several new infinite classes of (xvt,xvt−1)-minihypers in PG(t,q) that cannot be written as an integer sum of hyperplanes
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