In [Dempwolff :\textit{More Translation Planes and Semifields from Dembowski-Ostrom Polynomials}, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81--103], the author gives a construction of three classes of rank two semifields of order $q^{2n}$, 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 $\cD_{A}$ and $\cD_{AB}$, are new for $n>3$, whereas presemifields in family ${\cD}_{B}$ are isotopic to Generalized Twisted Fields for each $n\geq 3$.

Solution to an isotopism question concerning rank 2 semifields

MARINO, Giuseppe;POLVERINO, Olga;
2015

Abstract

In [Dempwolff :\textit{More Translation Planes and Semifields from Dembowski-Ostrom Polynomials}, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81--103], the author gives a construction of three classes of rank two semifields of order $q^{2n}$, 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 $\cD_{A}$ and $\cD_{AB}$, are new for $n>3$, whereas presemifields in family ${\cD}_{B}$ are isotopic to Generalized Twisted Fields for each $n\geq 3$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11591/235015
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