In \cite{Lunardon}, G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order $q^{2n},$ $n>1$ and $n$ odd, with left nucleus $\F_{q^n}$, middle and right nuclei both $\F_{q^2}$ and center $\F_q$. When $n=3$ this method gives an alternative construction of a family of semifields described in \cite{JMPT2}, which generalizes the family of cyclic semifields obtained by Jha and Johnson in \cite{jj}. For $n> 3$, no example of a semifield belonging to this family is known. In this paper we first prove that, when $n>3$, any semifield belonging to the family introduced in \cite{JMPT2} is not isotopic to any semifield of the family constructed in \cite{Lunardon}. Then we construct, with the aid of a computer, a semifield of order $2^{10}$ belonging to the family introduced by Lunardon, which turns out to be non--isotopic to any other known semifield.
A new semifield of order $2^{10}$
MARINO, Giuseppe;
2010
Abstract
In \cite{Lunardon}, G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order $q^{2n},$ $n>1$ and $n$ odd, with left nucleus $\F_{q^n}$, middle and right nuclei both $\F_{q^2}$ and center $\F_q$. When $n=3$ this method gives an alternative construction of a family of semifields described in \cite{JMPT2}, which generalizes the family of cyclic semifields obtained by Jha and Johnson in \cite{jj}. For $n> 3$, no example of a semifield belonging to this family is known. In this paper we first prove that, when $n>3$, any semifield belonging to the family introduced in \cite{JMPT2} is not isotopic to any semifield of the family constructed in \cite{Lunardon}. Then we construct, with the aid of a computer, a semifield of order $2^{10}$ belonging to the family introduced by Lunardon, which turns out to be non--isotopic to any other known semifield.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
