Solution to an isotopism question concerning rank 2 semifields

Original · EN

In [U. Dempwolff: More Translation Planes and Semifields from Dembowski-Ostrom Polynomials, Designs, Codes, Cryptogr. 68 (1-3) (2013), 81-103], the author gives a construction of three classes of rank two semifields of order q²ⁿ, with q and n odd, using Dembowski-Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for n>3, in particular we prove that two of these classes, labeled Dₐ and DAB, are new for n>3, whereas presemifields in family DB are isotopic to Generalized Twisted Fields for each n≥ 3.