On isotopisms and strong isotopisms of commutative presemifields

In this paper we prove that the $P(q,\mathrm{\ell })$ ($q$ odd prime power and $\mathrm{\ell }>1$ 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 $q^{2\mathrm{\ell }}$ ($\mathrm{\ell }>1$ odd) defining CCZ–inequivalent planar DO polynomials.