The semifields of order q6 which are two-dimensional over their left nucleus andsix-dimensional over their center have been geometrically partitioned into six classesby using the associated linear sets in PG(3, q3). One of these classes has been par-titioned further (again geometrically) into three subclasses. In this paper algebraiccurves are used to construct two infinite families of odd order semifields belongingto 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 inthis subclass which have the right or middle nucleus which is two-dimensional overthe center. This work is a non-trivial step towards the classification of all semifieldsthat are six-dimensional over their center and two-dimensional over their left nu-cleus.
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