A new semifield of order 2¹⁰

Abstract

In [G. Lunardon, Semifields and linear sets of $PG(1,q^t)$, Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], 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 ${\mathbb{F}}_{q^n}$, middle and right nuclei both ${\mathbb{F}}_{q^2}$ and center ${\mathbb{F}}_q$. When $n=3$ this method gives an alternative construction of a family of semifields described in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1–34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87–104]. 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 the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 2¹⁰ belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield.