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Switchings of semifield multiplications

Hou, Xiang-dong
Özbudak, Ferruh
Let B(X, Y) be a polynomial over F-qn which defines an F-q-bilinear form on the vector space F-qn, and let xi be a nonzero element in F-qn. In this paper, we consider for which B(X, Y), the binary operation xy + B(x, y) xi defines a (pre)semifield multiplication on F-qn. We prove that this question is equivalent to finding q-linearized polynomials L(X) is an element of F-qn [X] such that Tr-qn/q (L(x)/x) not equal 0 for all x is an element of F-qn*. For n <= 4, we present several families of L(X) and we investigate the derived (pre) semifields. When q equals a prime p, we show that if n > 1/2(p - 1)(p(2) - p + 4), L(X) must be a(0)X for some a(0) is an element of F-pn satisfying Tr-qn/q (a(0)) not equal 0. Finally, we include a natural connection with certain cyclic codes over finite fields, and we apply the Hasse-Weil-Serre bound for algebraic curves to prove several necessary conditions for such kind of L(X).