A Generalization of Some Huang–Johnson Semifields

Abstract

In [H. Huang, N.L. Johnson: Semifield planes of order $8^2$, Discrete Math., 80 (1990)], the authors exhibited seven sporadic semifields of order $2^6$, with left nucleus ${\mathbb F}_{2^3}$ and center ${\mathbb F}_2$. Following the notation of that paper, these examples are referred as the Huang–Johnson semifields of type $II$, $III$, $IV$, $V$, $VI$, $VII$ and $VIII$. In [N. L. Johnson, V. Jha, M. Biliotti: Handbook of Finite Translation Planes, Pure and Applied Mathematics, Taylor Books, 2007], the question whether these semifields are contained in larger families, rather then sporadic, is posed. In this paper, we first prove that the Huang–Johnson semifield of type $VI$ is isotopic to a cyclic semifield, whereas those of types $VII$ and $VIII$ belong to infinite families recently constructed in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti: Semifields of order $q^6$ with left nucleus ${\mathbb F}_{q^3}$ and center ${\mathbb F}_q$, Finite Fields Appl., 14 (2008)] and [G.L. Ebert, G. Marino, O. Polverino, R. Trombetti: Infinite families of new semifields, Combinatorica, 6 (2009)]. Then, Huang–Johnson semifields of type $II$ and $III$ are extended to new infinite families of semifields of order $q^6$, existing for every prime power $q$.