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

Authors:	Email author" target="_blank">Ferruh??zbudak Email author

Institution:	(1) Department of Mathematics, Middle East Technical University, nönü Bulvar, 06531 Ankara, Turkey

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.classification 11T30, 11G20, 05B15

Keywords:	finite field algebraic function field BCH code ordered orthogonal array
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