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  • 06 C 10  (1)
  • Flag-transitive  (1)
  • 1
    Electronic Resource
    Electronic Resource
    On the number of nowhere zero points in linear mappings (1994)
    Springer
    Combinatorica 14 (1994), S. 149-157 
    Details
    ISSN: 1439-6912
    Keywords: 06 C 10 ; 15 A 06 ; 11 T 99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract LetA be a nonsingularn byn matrix over the finite fieldGF q ,k=⌊n/2⌋,q=p a ,a≥1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF q ) n such that bothx andAx have no zero component. We prove that forn≥2, and $$q 〉 2\left( {\begin{array}{*{20}c} {2n} \\ 3 \\ \end{array} } \right)$$ ,P(A,q)≥[(q−1)(q−3)] k (q−2) n−2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)≥1, is true for allq≥n+2≥3 orq≥n+1≥4.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01215347
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  • 2
    Electronic Resource
    Electronic Resource
    Two-dimensional flag-transitive planes revisited (1996)
    Springer
    Geometriae dedicata 63 (1996), S. 1-15 
    Details
    ISSN: 1572-9168
    Keywords: 51A40 ; 51E15 ; Flag-transitive ; translation plane ; two-dimensional ; enumeration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q 2 is precisely (q−1)/2 when q is an odd prime and precisely (q−1)/2e when q=p e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00181182
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