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Keywords:

  • relative difference set;
  • commutative semifield;
  • projective plane;
  • isotopism

Abstract

We show that every inline image-relative difference set D in inline image relative to inline image can be represented by a polynomial inline image, where inline image is a permutation for each nonzero a. We call such an f a planar function on inline image. The projective plane Π obtained from D in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on inline image with exactly two elements in its image set and inline image is planar, if and only if, inline image for any inline image.