On semifields of type (q^{2n},q^n,q^2,q^2,q), n odd

We investigate semifields of order $q^{2n}$, n odd, having left nucleus of order $q^n$, middle and right nucleus both of order $q^2$, and center of order q. Let $U, V$ be complementary $F_{q^2}$ -subspaces of $F_{q^{2n}}$ such that $xx^{q^n} \neq yy^{q^n}$ for all $x \in U \setminus {0},$ $y \in V \setminus {0}.$ Let $P:F_q^{2n} \mapsto F_q^{2n}$ be the projection with kernel U and image V, and define a multiplication◦: $Fq2n ×F_q^{2n} \mapsto F_q^{2n}$ by x \circ y=xy+(xq^n −x)P(y).$ Then $(F_{q^n},+,\circ)$ is a semifield of the type required, and, up to isotopy, any such semifield can be obtained in this way. One may assume that the rank of V is less than or equal to the rank of U , and under this assumption the rank j of U is an isotopy invariant. Examples with $j = \frac{n+1}{2}$ are known for any value of q and n. The authors construct four new mutually nonisotopic examples with q = 2, n = 7, j = 5.