On isotopisms and strong isotopisms of commutative presemifields

In this paper we prove that the P(q,l) (q odd prime power and l>1 odd) commutative semifields constructed by Bierbrauer (Des. Codes Cryptogr. 61:187-196, 2011) are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth (SETA, pp. 403-414, 2008). Also, we show that they are strongly isotopic if and only if q\congr 1(mod 4). Consequently, for each q\congr 1(mod 4) there exist isotopic commutative presemifields of order q^{2l} (l>1 odd) defining CCZ-inequivalent planar DO polynomials