Any finite semifield $2$--dimensional over its left nucleus and $2n$--dimensional over its center defines a linear set of rank $2n$ of $PG(3,q^n)$ disjoint from a hyperbolic quadric and conversely. Using this connection, semifields 2--dimensional over their left nucleus and 4--dimensional over their center were classified. In this paper we give a characterization result in the case $n=3$, proving that there exist five or six non--isotopic families of such semifields, the families $\cF_i$, $i=0,\dots,5$ ($\cF_3$ might be empty), according to the different configurations of the associated linear sets of $PG(3,q^3)$. Also, we prove that to any semifield belonging to the family $\cF_5$ is associated an $\F_q$--{\it pseudoregulus} of $PG(3,q^3)$ and we characterize the known examples of semifields of the family $\cF_5$ in terms of the associated $\F_q$--pseudoregulus.
On $F_ q$--linear sets of $PG(3,q^3)$ and semifields
POLVERINO, Olga;MARINO, Giuseppe;
2007
Abstract
Any finite semifield $2$--dimensional over its left nucleus and $2n$--dimensional over its center defines a linear set of rank $2n$ of $PG(3,q^n)$ disjoint from a hyperbolic quadric and conversely. Using this connection, semifields 2--dimensional over their left nucleus and 4--dimensional over their center were classified. In this paper we give a characterization result in the case $n=3$, proving that there exist five or six non--isotopic families of such semifields, the families $\cF_i$, $i=0,\dots,5$ ($\cF_3$ might be empty), according to the different configurations of the associated linear sets of $PG(3,q^3)$. Also, we prove that to any semifield belonging to the family $\cF_5$ is associated an $\F_q$--{\it pseudoregulus} of $PG(3,q^3)$ and we characterize the known examples of semifields of the family $\cF_5$ in terms of the associated $\F_q$--pseudoregulus.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
