On the multiplication of some semifields of order q6

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 into three subclasses. In this paper the generic multiplication is determined for each of these three subclasses, and several examples of new semifields are constructed that belong to these subclasses. For two of the subclasses, no examples were previously known.