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 $T{r}_{q^n/q}(L(x)/x)\ne 0$ for all $x\in {\mathbb{F}}_{q^n}^{\ast }$. 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 $T{r}_{q^n/q}(a_0)\ne 0$. 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 $L(X)$.

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