Semifields in Class F4(a)

The semifields of order q6 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, q3). 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. This author acknowledges the support of NSA grant H98230-09-1-0074 This work was supported by the Research Project of MIUR (Italian Office for University and Research) “Strutture geometriche, combinatoria e loro applicazioni”and by the Research group GNSAGA of INDAM the electronic journal of combinatorics 16 (2009), #R53 1