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1 results

  • ON A DIVISIBILITY PROBLEM FOR POLYNOMIALS AND ITS APPLICATION TO CAMERON–PRAEGER DESIGNS

  • NGO DAC TUAN
  • Journal:
    Journal of the London Mathematical Society / Volume 67 / Issue 3 / June 2003
    Published online by Cambridge University Press:
    20 May 2003, pp. 545-560
    Print publication:
    June 2003

  • Cameron–Praeger designs with parameters $t-(v,k,\lambda)$ are studied. Cameron and Praeger showed that in such designs, $t=2$ or 3. In 1989, Delandtsheer and Doyen proved that if $t=2$ then \[ v \leqslant \left(\left({k \atop 2}\right)-1\right)^2. \] In 2000, Mann and Tuan improved this equality and showed that if $t=3$ then \[ v \leqslant \left({k \atop 2}\right)+1. \] Three infinite families of Cameron–Praeger 3-designs for which this bound is met have been constructed by Mann and Tuan and by Sebille. The paper constructs infinitely many infinite families of such Cameron–Praeger 3-designs via a study of a divisibility problem for polynomials. Further, the construction generalizes the previous constructions.