Generalized $j$-planes

Abstract

We construct and study a class of translation planes with kernel $K\cong GF(q)$, order $q^n$, and $n>2$. These planes generalize the $j$-planes discovered by Johnson, Pomareda and Wilke in [14]. We show these planes are actually $jj \cdots j$-planes. Hence, most of the results obtained in this article are on $jj\cdots j$-planes. In fact, our study shows that these planes are either nearfield or new.

An infinite class of nearfield $jj\cdots j$-planes is shown to exist, and a finite set of sporadic non-Andr\'e $jj\cdots j$-planes is presented.