On isotopisms and strong isotopisms of commutative presemifields

Abstract

In this paper we prove that the $P(q,\ell )$ ($q$ odd prime power and $\ell >1$ odd) commutative semifields constructed by Bierbrauer in \cite{BierbrauerSub} are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth in \cite{BuHe2008}. Also, we show that they are strongly isotopic if and only if $q\equiv 1(mod4)$. Consequently, for each $q\equiv -1(mod4)$ there exist isotopic commutative presemifields of order $q^{2\ell }$ ($\ell >1$ odd) defining CCZ--inequivalent planar DO polynomials.

References updated, pag. 5 corrected Multiplication of commutative LMPTB semifields