The six semifield planes associated with a semifield flock

Abstract In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array ( a ijk ) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions ( f,g) where f and g are functions from GF ( q ) to GF ( q ) with the property that f and g are linear over some subfield and g ( x ) 2 +4 xf ( x ) is a non-square for all x∈GF(q) ∗ , q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q (4, q ). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless ( f,g) are of linear type or of Dickson–Kantor–Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.

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