Switchings of semifield multiplications

Abstract

Let $B(X,Y)$ be a polynomial over $\mathbb {F}_{q^n}$ which defines an $\mathbb {F}_q$-bilinear form on the vector space $\mathbb {F}_{q^n}$, and let $\xi$ be a nonzero element in $\mathbb {F}_{q^n}$. In this paper, we consider for which $B(X,Y)$, the binary operation $xy+B(x,y)\xi$ defines a (pre)semifield multiplication on ${\mathbb {F}}_{q^n}$. We prove that this question is equivalent to finding $q$-linearized polynomials $L(X)\in \mathbb {F}_{q^n}[X]$ such that ${\mathrm {Tr}}_{q^n/q}(L(x)/x)\ne 0$ for all $x\in \mathbb {F}_{q^n}^*$. For $n\le 4$, we present several families of $L(X)$ and we investigate the derived (pre)semifields. When $q$ equals a prime $p$, we show that if $n>\frac{1}{2}(p-1)(p^2-p+4)$, $L(X)$ must be $a_0 X$ for some $a_0\in \mathbb {F}_{p^n}$ satisfying ${\mathrm {Tr}}_{q^n/q}(a_0)\ne 0$. 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 $L(X)$.