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Rédei Blocking Sets in Finite Desarguesian Planes

2002
*
Journal of combinatorial theory. Series A
*

PRELIMINARIES A

doi:10.1006/jcta.2001.3242
fatcat:34ugzhcapbepzkfkbnqsnembya
*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. ...##
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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. ...##
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Page 5593 of Mathematical Reviews Vol. , Issue 97I
[page]

1997
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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. ...##
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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’. ...##
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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*. ...

##
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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. ...

##
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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]

2014
*
arXiv
*
pre-print

This paper is a survey paper on old and recent results on direction problems

arXiv:1409.6960v1
fatcat:5x4wlbtqifh7re3mhhmrwdfl5e
*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*. ...##
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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). ...##
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Page 3260 of Mathematical Reviews Vol. , Issue 99e
[page]

1999
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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}. ...##
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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. ...##
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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. ...##
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Blocking sets and semifields

2006
*
Journal of combinatorial theory. Series A
*

We find a relationship between semifield spreads of PG(3, q), small

doi:10.1016/j.jcta.2005.11.001
fatcat:yl4aoqf47jbtjloifhabd3kmmm
*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. ...##
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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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