The two sets of three semifields associated with a semifield flock

Abstract

In 1965 Knuth showed that from a given finite semifield one can construct further semifields manipulating the corresponding cubical array, and obtain in total six semifields from the given one. In the case of a rank two commutative semifield (the semifields corresponding to a semifield flock) these semifields have been investigated by Ball and Brown (2004), providing a geometric connection between these six semifields and it was shown that they give at most three non-isotopic semifields. However, there is another set of three semifields arising in a different way from a semifield flock, hence in total six semifields arise from a rank two commutative semifield. In this article we give a geometrical link between these two sets of three semifields.