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Rédei Blocking Sets in Finite Desarguesian Planes

B.F. Sherman
2002 Journal of combinatorial theory. Series A  
PRELIMINARIES A blocking set in a projective plane is a set of points of the plane which contains at least one point of each line, but not all the points of any of them.  ...  Sherman [7] has characterised Rédei blocking sets with at least two Rédei lines in terms of additive and multiplicative groups of GF(q).  ...  projective plane of order q which meets at least one line in exactly n points-and Bruen and Silverman [2] renamed this concept a blocking set of Rédei type.  ... 
doi:10.1006/jcta.2001.3242 fatcat:34ugzhcapbepzkfkbnqsnembya

Page 5622 of Mathematical Reviews Vol. , Issue 2001H [page]

2001 Mathematical Reviews  
The blocking sets of Rédei type in PG(2, p*) have size p> + p? + 1 or p> + p? + p+, as was proved by the author earlier. For g = p, p prime, there are no small blocking sets [A.  ...  The very nice paper under review shows that all small minimal blocking sets in PG(2,q), gq = p*, p prime, p >7, are of Rédei type.  ... 

Page 5593 of Mathematical Reviews Vol. , Issue 97I [page]

1997 Mathematical Reviews  
A connection of U with an inversive plane, known in the case where I is Desarguesian, is extended to the non- Desarguesian case.  ...  A blocking set B is a set of points intersecting every line but not containing a line. It is said to be of Rédei type if there is a line intersecting it in exactly |B| - q points.  ... 

Page 7387 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
When every line meets a blocking set B in | modulo gq points then B is either a Baer subplane (which exists only when q is a square) or B is of Rédei type.  ...  Codes Cryptogr. 20 (2000), no. 3, 319-324; MR 2001h:51018] proved that minimal blocking sets of size less than 3(g + 1)/2 are of Rédei type when g = p’.  ... 

Page 6903 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
In a Rédei blocking set, a subset of / collinear points is called a Rédei line. In this paper, Rédei blocking sets with at least two Rédei lines and 3 <h <gq-—1 are examined.  ...  is called a Rédei blocking set.  ... 

Page 4037 of Mathematical Reviews Vol. , Issue 2001F [page]

2001 Mathematical Reviews  
In the Desarguesian plane PG(2, q‘) one can construct a blocking set of Rédei type, having q‘ + q‘~'+1 points, starting from the graph of the trace function from GF(q‘) to GF(q) [see A.  ...  B is called a blocking set of Rédei type, if there is a line / for which |BN/| =|B|— q. In this case the line / is called a Rédei line for B.  ... 

Page 3586 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
2003e:5 1015 sizes of small minimal blocking sets: the Rédei type GF(q)-linear blocking sets must have size g? + q°>+1 and g++q°+q?  ...  +cq+1, c € {—1,0,1}, while the non-Rédei type blocking sets must have size gi + q3+q°?+dq+1, with d € {0,1}. It is also shown that each possible cardinality actually occurs.  ... 

Direction problems in affine spaces [article]

Jan De Beule
2014 arXiv   pre-print
This paper is a survey paper on old and recent results on direction problems in finite dimensional affine spaces over a finite field.  ...  More information on blocking sets of Desarguesian projective planes can be found in [17] . The situation for non-Desarguesian planes is more complicated, especially the construction of examples.  ...  Apart from [18] and [19] , the papers [10, 11, 30] are interesting references for blocking sets of non Desarguesian planes.  ... 
arXiv:1409.6960v1 fatcat:5x4wlbtqifh7re3mhhmrwdfl5e

Page 5039 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
Linear blocking sets pro- vided the first examples of small minimal blocking sets which are not of Rédei type.  ...  In the nice paper under review the authors show that a GF(q)- linear blocking set of Rédei type defines a derivable partial spread in PG(2t —1,q).  ... 

Page 3260 of Mathematical Reviews Vol. , Issue 99e [page]

1999 Mathematical Reviews  
A blocking set is said to be of Rédei type if there is a line / such that |@ \/| = q; these are extremal in the sense that for any blocking set # and line / in z we have |#~/| > q.  ...  In this case / is called the Rédei line. Further, a blocking set @ in PG(2, p) is said to be of almost Rédei type if there is a line / such that |A \/| € {p,p+1,p+2}.  ... 

Page 1253 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
The authors construct small minimal blocking sets not of Rédei type in an infinite family of non-Desarguesian translation planes, namely, André planes constructed from spreads in projective spaces.  ...  A (nontrivial) blocking set in a projective plane I is a set @ of points such that every line meets (but is not contained in) #. & is minimal if no proper subset is a blocking set.  ... 

Page 6198 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews  
There is only one known minimal blocking set B in PG(2, p) that is not of Rédei type.  ...  satisfies p <7 and all such blocking sets are of Rédei type (relative to a line #/) apart from the exceptional example in PG(2,7) indicated above.  ... 

Page 1370 of Mathematical Reviews Vol. , Issue 87c [page]

1987 Mathematical Reviews  
The methods in both papers are nonconstructive. S. Stein (Davis, Calif.) 87c:51012 Blokhuis, A. (NL-EIND); Brouwer, A. E. (NL-MATH) Blocking sets in Desarguesian projective planes. Bull. London Math.  ...  A blocking set in a projective plane P of order q is a set B of points such that any line contains a point of B and a point off B. It is well known [A. A. Bruen, SIAM J. Appl.  ... 

Blocking sets and semifields

Guglielmo Lunardon
2006 Journal of combinatorial theory. Series A  
We find a relationship between semifield spreads of PG(3, q), small Rédei minimal blocking sets of PG(2, q 2 ), disjoint from a Baer subline of a Rédei line, and translation ovoids of the hermitian surface  ...  I gratefully acknowledge the referee for having pointed out that in [17] Johnson has proved that a transversal of a derivable net produces an indicator set, and therefore transversal functions and indicator  ...  sets are equivalent; hence, the results and the techniques of [17] might be used to provide an answer to some of the problems posed here.  ... 
doi:10.1016/j.jcta.2005.11.001 fatcat:yl4aoqf47jbtjloifhabd3kmmm

Page 1471 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews  
A blocking set in a projective plane is a set B of points such that any line has a point in B as well as a point outside B. If the projective plane has order n = m?  ...  Kallaher (1-WAS) 92¢:51015 51E21 05B25 Drake, David A. (1-FL) A bound for blocking sets in finite projective planes. Finite geometries and combinatorial designs (Lincoln, NE, 1987), 93-97, Contemp.  ... 
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