A Generalization of Some Huang–Johnson Semifields

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 $q$.

SOME CHARACTERIZATIONS OF DESARGUESIAN TRANSLATION PLANES

In this note we consider finite translation planes with large translation complements. In particular, we characterize finite affine Desarguesian translation planes in two ways, according to the existence of subgroups in the translation complement that are divisible by relatively large integers, together with modest additional restrictions.