Abstract

A semifield of type $\left(q^{2n},q^n,q^2,q^2,q\right)$ (with $n>1$) is a finite semifield of order $q^{2n}$ ($q$ a prime power) with left nucleus of order $q^n$, right and middle nuclei both of order $q^2$ and center of order $q$. Semifields of type $\left(q^6,q^3,q^2,q^2,q\right)$ have been completely classified by the authors and N. L. Johnson. In this paper we determine, up to isotopy, the form of any semifield of type $\left(q^{2n},q^n,q^2,q^2,q\right)$ when $n$ is an odd integer, proving that there exist $\frac{n-1}{2}$ non isotopic potential families of semifields of this type. Also, we provide, with the aid of the computer, new examples of semifields of type $\left(q^{14},q^7,q^2,q^2,q\right)$, when $q=2$.

Keywords

Mathematical Subject Classification 2000

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