Elements of Prescribed Order, Prescribed Traces and Systems of Rational Functions Over Finite Fields

abstract

Let k ≥ 1 and $f{_1}, \ldots, f{_r} \in {\mathbb F}_{q^k}(x)$be a system of rational functions forming a strongly linearly independent set over a finite field ${\mathbb F}_q$. Let $\gamma_1, \ldots, \gamma_r \in {\mathbb F}_q$ be arbitrarily prescribed elements. We prove that for all sufficiently large extensions ${\mathbb F}_{q^{km}}$, there is an element $\xi \in {\mathbb F}_{q^{km}}$ of prescribed order such that ${\rm Tr}_{{\mathbb F}_{q^{km} }/{\mathbb F}_q}(f_i(\xi))=\gamma_i$ for $i=1, \ldots, r$, where ${\rm Tr}_{{\mathbb F}_{q^{km}/{\mathbb F}_q}}$ is the relative trace map from ${\mathbb F}_{q^{km}}$ onto \({\mathbb F}_q$. We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.