On isotopisms and strong isotopisms of commutative presemifields
Copyright policy )On isotopisms and strong isotopisms of commutative presemifields
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.
References updated, pag. 5 corrected Multiplication of commutative LMPTB semifields
Commutative semifields; Isotopy; Planar DO polynomials; Strong isotopy; Symplectic semifields; Algebra and Number Theory; Discrete Mathematics and Combinatorics, Algebra and Number Theory, Isotopy, Mathematics - Commutative Algebra, Commutative Algebra (math.AC), Commutative semifields; Isotopy; Planar DO polynomials; Strong isotopy; Symplectic semifields, Strong isotopy, Symplectic semifields, FOS: Mathematics, Planar DO polynomials, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Commutative semifields, Combinatorics (math.CO)
Commutative semifields; Isotopy; Planar DO polynomials; Strong isotopy; Symplectic semifields; Algebra and Number Theory; Discrete Mathematics and Combinatorics, Algebra and Number Theory, Isotopy, Mathematics - Commutative Algebra, Commutative Algebra (math.AC), Commutative semifields; Isotopy; Planar DO polynomials; Strong isotopy; Symplectic semifields, Strong isotopy, Symplectic semifields, FOS: Mathematics, Planar DO polynomials, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Commutative semifields, Combinatorics (math.CO)
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