Elements of prescribed order, prescribed traces and systems of rational functions over finite fields

Ozbudak F.

DESIGNS CODES AND CRYPTOGRAPHY, vol.34, no.1, pp.35-54, 2005 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 34 Issue: 1
  • Publication Date: 2005
  • Doi Number: 10.1007/s10623-003-4193-0
  • Journal Name: DESIGNS CODES AND CRYPTOGRAPHY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.35-54
  • Middle East Technical University Affiliated: Yes

Abstract

Let k greater than or equal to 1 and f(1),..., f(r) is an element of F-qk (x) be a system of rational functions forming a strongly linearly independent set over a finite field F-q. Let gamma(1),..., gamma(r) is an element of F-q be arbitrarily prescribed elements. We prove that for all sufficiently large extensions F-qkm, there is an element xi is an element of F-qkm of prescribed order such that Tr-Fqkm /Fq (f(i) (xi)) = gamma(i) for i = 1,..., r, where Tr-Fqkm/fq is the relative trace map from F-qkm onto 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.