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  • Regulus  (2)
  • Baer subplane  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Regulus-free Spreads of PG(3,q) (1996)
    Springer
    Designs, codes and cryptography 8 (1996), S. 79-89 
    Details
    ISSN: 1573-7586
    Keywords: Regulus ; spread ; flag-transitive
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An old conjecture of Bruck and Bose is that every spreadof Σ = PG(3,q) could be obtained by startingwith a regular spread and reversing reguli. Although it was quicklyrealized that this conjecture is false, at least for qeven, there still remains a gap in the spaces for which it isknown that there are spreads which are regulus-free. In severalpapers Denniston, Bruen, and Bruen and Hirschfeld constructedspreads which were regulus–free, but none of these dealtwith the case when q is a prime congruent to onemodulo three. This paper closes that gap by showing that forany odd prime power q, spreads of PG(3,q) yielding nondesarguesian flag-transitive planes are regulus–free.The arguments are interesting in that they are based on elementarylinear algebra and the arithmetic of finite fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018024723184
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Perfect Baer Subplane Partitions and Three-Dimensional Flag-Transitive Planes (2000)
    Springer
    Designs, codes and cryptography 21 (2000), S. 19-39 
    Details
    ISSN: 1573-7586
    Keywords: Baer subplane ; flag-transitive affine plane ; linearized polynomials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The classification of perfectBaer subplane partitions of PG(2, q2) is equivalentto the classification of 3-dimensional flag-transitive planeswhose translation complements contain a linear cyclic group actingregularly on the line at infinity. Since all known flag-transitiveplanes admit a translation complement containing a linear cyclicsubgroup which either acts regularly on the points of the lineat infinity or has two orbits of equal size on these points,such a classification would be a significant step towards theclassification of all 3-dimensional flag-transitive planes. Usinglinearized polynomials, a parametric enumeration of all perfectBaer subplane partitions for odd q is described.Moreover, a cyclotomic conjecture is given, verified by computerfor odd prime powers q 〈 200, whose truth would implythat all perfect Baer subplane partitions arise from a constructionof Kantor and hence the corresponding flag-transitive planesare all known.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008319107762
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Regulus-free spreads ofPG (3,q) (1996)
    Springer
    Designs, codes and cryptography 8 (1996), S. 79-89 
    Details
    ISSN: 1573-7586
    Keywords: Regulus ; spread ; flag-transitive
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An old conjecture of Bruck and Bose is that every spread of Σ =PG(3,q) could be obtained by starting with a regular spread and reversing reguli. Although it was quickly realized that this conjecture is false, at least forq even, there still remains a gap in the spaces for which it is known that there are spreads which are regulus-free. In several papers Denniston, Bruen, and Bruen and Hirschfeld constructed spreads which were regulus-free, but none of these dealt with the case whenq is a prime congruent to one modulo three. This paper closes that gap by showing that for any odd prime powerq, spreads ofPG(3,q) yielding nondesarguesian flag-transitive planes are regulus-free. The arguments are interesting in that they are based on elementary linear algebra and the arithmetic of finite fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00130569
    Location Call Number Expected Availability
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    Paper (German National Licenses)
    Fulltext
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