On isotopisms and strong isotopisms of commutative presemifields

TL;DR

This paper investigates the isotopism properties of certain commutative semifields, establishing conditions for strong isotopism and revealing the existence of CCZ-inequivalent planar polynomials for specific parameters.

Contribution

It proves isotopy between specific classes of commutative semifields and characterizes when they are strongly isotopic, highlighting new distinctions in their algebraic structure.

Findings

P(q,ℓ) semifields are isotopic to those by Budaghyan and Helleseth.

Strong isotopism occurs iff q ≡ 1 mod 4.

Existence of CCZ-inequivalent planar DO polynomials for q ≡ -1 mod 4.

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 $q\equiv 1(mod\,4)$. Consequently, for each $q\equiv -1(mod\,4)$ there exist isotopic commutative presemifields of order $q^{2\ell}$ ($\ell>1$ odd) defining CCZ--inequivalent planar DO polynomials.