Primitive elements and polynomials with arbitrary trace
Discrete Mathematics
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Hypercubic 4 and 5-Designs from Double-Error-Correcting BCH Codes
Designs, Codes and Cryptography
Primitive Elements with Zero Traces
Finite Fields and Their Applications
Finite Field Elements with Specified Order and Traces
Designs, Codes and Cryptography
Improvements on Generalized Hamming Weights of Some Trace Codes
Designs, Codes and Cryptography
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Let k ≥ 1 and f1 .... , fr ∈ Fqk (x) be a system of rational functions forming a strongly linearly independent set over a finite field Fq. Let γ1 ..... γr ∈ Fq be arbitrarily prescribed elements. We prove that for all sufficiently large extensions Fqkm, there is an element ξ ∈ Fqkm of prescribed order such that TrFqkm/Fq (fi (ξ))= γi for i = 1, ..., r, where TrFqkm/Fq is the relative trace map from Fqkm onto Fq. 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.