\(j,k\)-planes of order \(4^3\) (Q2493211)

In the paper a new class of translation planes of order \(4^3\) is constructed and studied. In 1992 the first author constructed an affine plane of order \(4^3\) and kernel GF\((4)\) admitting an affine homology group of order \((4^3 - 1)/(4 - 1)\), which, in fact, is not André. The third author, using the computer, was able to construct a large set of such planes, and the paper under review extends both of these two constructions and is part of the second author's PhD thesis at the University of Iowa. The idea of the basic constructions involves extending the definition of \(j\)--planes for planes of order \(q^2\), to \(j\cdots j\)-planes for planes of order \(q^n\). When \(n=3\), these planes are called \(jj\)-planes, or \(j, k\)-planes. First, the following theorem is proved. Theorem. There are three isomorphism classes of \(j, k\)-planes of order \(4^3\). \noindent One of these planes is a nearfield plane and the other two are new planes. All such planes have kernel GF\((4)\) and spreads in PG\((5, 4)\). \noindent Each \(j, k\)-plane admits a collineation group \(G\) of order \(4^3 - 1\) fixing two components and transitive on the remaining components. \noindent Within \(G\), there is an affine homology group of order \((4^3 - 1)/(4 - 1)\) producing three nets \((\)André nets\()\) of the same size that are replaceable by two distinct replacements. Then, by replacing the nets, new planes are constructed and studied.