Semifields in Class ${\cal F}_4^{(a)}$

  • Gary Ebert
  • Giuseppe Marino
  • Olga Polverino
  • Rocco Trombetti


The semifields of order $q^6$ which are two-dimensional over their left nucleus and six-dimensional over their center have been geometrically partitioned into six classes by using the associated linear sets in $PG(3,q^3)$. One of these classes has been partitioned further (again geometrically) into three subclasses. In this paper algebraic curves are used to construct two infinite families of odd order semifields belonging to one of these subclasses, the first such families shown to exist in this subclass. Moreover, using similar techniques it is shown that these are the only semifields in this subclass which have the right or middle nucleus which is two-dimensional over the center. This work is a non-trivial step towards the classification of all semifields that are six-dimensional over their center and two-dimensional over their left nucleus.

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